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High Strain Rate Characterisation of Composite Materials
Eriksen, Rasmus Normann Wilken
Publication date:2014
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Citation (APA):Eriksen, R. N. W. (2014). High Strain Rate Characterisation of Composite Materials. DTU MechanicalEngineering. DCAMM Special Report, No. S179
Ph
D T
he
sis
High Strain Rate characterisation of Composite materials
Rasmus Normann Wilken EriksenDCAMM Special Report No. S179March 2014
High Strain Rate characterisation of Composite materials
Rasmus Normann Wilken Eriksen 31/03/2014
Department of Mechanical Engineering
I
Table of contents
1 Introduction ........................................................................................................................................................................1 1.1 Thesis structure and novelty ................................................................................................................. 3 1.2 Strain .................................................................................................................................................... 4 1.3 Strain rate.............................................................................................................................................. 5 1.4 Dynamic deformation ........................................................................................................................... 6 1.5 Strain rate sensitivity of FRP materials ............................................................................................... 10
2 Assessment of servo hydraulic test machine as high speed loading device...................................................................14 2.1 Description of the high servo hydraulic machine................................................................................ 14 2.2 Numerical model ................................................................................................................................ 19 2.3 Verification of models ........................................................................................................................ 23 2.4 Examination of stress and strain rate response ................................................................................... 26 2.5 Summary ............................................................................................................................................ 33
3 Design and construction of a High- Speed Servo-Hydraulic Test Machine ................................................................35 3.1 Design ................................................................................................................................................. 35 3.2 Frame and load unit ............................................................................................................................ 36 3.3 Control and DAQ system .................................................................................................................... 40 3.4 Control electronics .............................................................................................................................. 43 3.5 Sensors ................................................................................................................................................ 43 3.6 Specimen design ................................................................................................................................. 43 3.7 Load train............................................................................................................................................ 45 3.8 Issue with grips ................................................................................................................................... 47 3.9 Summary ............................................................................................................................................ 48
4 Characterisation of fibre reinforced materials at medium strain rates .......................................................................49 4.1 Method ................................................................................................................................................ 49 4.2 Results ................................................................................................................................................ 51 4.3 Discussion........................................................................................................................................... 56 4.4 Summary ............................................................................................................................................ 57
5 Introduction to the Split Hopkinson Pressure Bar ........................................................................................................58 5.1 The Split Hopkinson Pressure Bar ...................................................................................................... 58 5.2 Pulseshaping ....................................................................................................................................... 64 5.3 Calibration of the SHPB setup ............................................................................................................ 70 5.4 Time synchronisation.......................................................................................................................... 74 5.5 Dispersion correction .......................................................................................................................... 75 5.6 Summary ............................................................................................................................................ 79
6 A generalized wave mechanic model of the Split Hopkinson Pressure Bar .................................................................80 6.1 Introduction ........................................................................................................................................ 80 6.2 1D generalised wave mechanic model ................................................................................................ 82 6.3 Validation of the model ...................................................................................................................... 88 6.4 Effect of the Incident wave ................................................................................................................. 90 6.5 Stress equilibrium and constant strain rate.......................................................................................... 92 6.6 Test design .......................................................................................................................................... 94 6.7 Radial inertia - considerations ............................................................................................................ 97 6.8 Simulation tool ................................................................................................................................... 97 6.9 Summary .......................................................................................................................................... 100
7 An alternative momentum trap for the Split Hopkinson Pressure Bar ..................................................................... 102 7.1 New method of momentum trapping ................................................................................................ 103 7.2 Summary .......................................................................................................................................... 107
8 Comparison of the through Thickness strain rate sensitivity of Eglass/Epoxy and Eglass/Lpet UD composite ..... 109 8.1 Materials ........................................................................................................................................... 109 8.2 Test setup .......................................................................................................................................... 110 8.3 Results .............................................................................................................................................. 112 8.4 Digital Image Correlation Check ...................................................................................................... 116 8.5 Summary .......................................................................................................................................... 118
Department of Mechanical Engineering
II
9 Summary and Conclusion ............................................................................................................................................. 119 9.1 Perspective ........................................................................................................................................ 121
Department of Mechanical Engineering
III
Preface
This thesis is submitted as a partial fulfilment of the requirements for the Danish Ph.D. degree.
The work was performed in the Structural Light Weights Group, Department of Mechanical
Engineering, Technical University of Denmark, during the period of August 2010 to March 2014.
The project was supervised by Associate Professor Christian Berggreen and co-supervised by
Professor Janice Barton, University of Southampton, and Senior Researcher Helmuth Toftegaard,
DTU Wind energy.
The study was part of an overall project, RESIST, which involved the participation of the Danish
Army, Falck Schmidt Defence Systems, COMFIL® and DTU Wind Energy. The RESIST project
encompassed another Ph.D. concerning blast testing and simulation of fibre reinforced polymer
composite panels. Test data obtained throughout this work was used as input data for the
simulation of the composite panels.
Department of Mechanical Engineering
IV
Acknowledgement
I would like to thank my Co-Supervisor Professor Janice Barton for hosting me at her university
during my external stay and for her motivating inputs. I would also like to thank my Co-
Supervisor Helmuth Toftegaard for many fruitful discussions and his help of organising test
specimens at the DTU Wind energy fibre lab. In addition, a great thanks to the skilled technician at
the DTU Wind energy fibre lab who produced all the test specimens with great care through the
project.
There is a very special thanks to Robert Swan, technician in building 119. Without him, it would
have been impossible to construct a complete test machine and install it in the lab! The same hold
for Jonathan Schwartz for his extraordinary time investment in designing and installing the pump
that pressurised the test machine. There is also a special thanks to all the skilled technician in the
workshop in building 119, building 413, and building 427, who made this very heavy experimental
project possible!
… And my very good friend and colleague Søren Giversen needs a special thanks for his endless
support, funny moment and fruitful inputs!
Special thanks to my supervisor Christian Berggreen who supported me through the hard times of
the project and encouraged me go on. A special thanks to Stina Jensen for her support and time
through the first part of the project.
…And of course the girlfriend! – Thanks to my lovely girlfriend Josefine for her patience – always
understanding when I was working in the night times, in the evenings, and when a good idea pop
up and I disappeared for the next three day to pursue my idea…
Østerbro 2014
Rasmus
Department of Mechanical Engineering
V
Abstract
The high strain rate characterisation of FRP materials present the experimenter with a new set of
challenges in obtaining valid experimental data. These challenges were addressed in this work
with basis in classic wave theory. The stress equilibrium process for linear elastic materials, as
fibre-reinforced polymers, were considered, and it was first shown that the loading history controls
equilibrium process. Then the High-speed servo-hydraulic test machine was analysed in terms its
ability to create a state of constant strain rate in the specimen. The invertible inertial forces in the
load train prevented a linear elastic specimen to reach a state of constant strain rate before fracture.
This was in contrast to ductile materials, which are widely tested with for the High-speed servo-
hydraulic test machine. The development of the analysis and the interpretation of the results, were
based on the experience from designing and constructing a high-speed servo-hydraulic test
machine and by performing a comprehensive test series. The difficulties encountered in the test
work could be addressed with the developed analysis. The conclusion was that the High-speed
servo-hydraulic test machine is less suited for testing fibre-reinforced polymers due to their elastic
behaviour and low strain to failure. This is problematic as the High-speed servo-hydraulic test
machine closes the gap between quasi-static tests rates and lower strain rates, which are achievable
with the Split Hopkinson Pressure Bar.
The Split Hopkinson Pressure Bar was addressed in terms of a new wave mechanics model for a
linear elastic specimen in the Split Hopkinson Pressure Bar. The model was formulated without
any assumption of stress equilibrium, constant strain rate, or equal bars and thus provided a useful
tool to analyse the equilibrium process. The model showed that whichever stress equilibrium of
constant strain rate happen first, depended on the combination of impedance mismatch between
the specimen and the bars. The model was compared to a series of tests, and the model correctly
indicated when a test set-up was problematic in terms of reaching stress equilibrium and constant
strain rate. As shown in literature the incident wave should be linear rising pulse to facilitate stress
equilibrium and constant strain rate. The common pulse shaping technique with copper disc’s
between the Striker bar and Incident bar were addresses and was concluded the method could
create the required Incident waves. However, there was an upper limit in the generated stress rates
due to frictional problems and this limited the maximum achievable strain rates. The maximum
strain rate was also found to be independent of the specimen gage length, which only controlled
the time to maximum strain rate.
The Split Hopkinson Pressure Bar proved able to reach a state of stress equilibrium and constant
strain rate, but the key to valid data was found in the control of the Incident wave.
Department of Mechanical Engineering
VI
Abstrakt
Materiale karakterisering af fiberforstærket plast materialer ved høje tøjningshastigheder er
udfordrende på grund af de hurtige belastninger, og for at opnå valide data skal testen være
designet så prøveemnet opnår en homogen deformation- og spændingstilstand inden brud. I dette
arbejde blev problem stillingen først beskrevet ved den klassiske bølgemekanik for elastiske
emner. Det blev vist at deformation hastigheden og accelerationen en er styrende for om emnet
opnår en homogen deformations og spændingstilstand inden brud. Udfordringerne var derfor at
styre deformationen af emnet korrekt. Højhastighedstrækprøvemaskinen har åbnet op for mulighed
for at testes ved middel tøjningshastigheder og dens evner til at opnå en homogen deformations og
spændingstilstand blev analyseret. Det blev fundet at inertikræfterne i last toget på maskinen
forhindrede emnet i at opnå en konstant tøjningshastighed før brud. Dette blev verificeret via en
række forsøg med en højhastighedstrækprøvemaskine der blev udviklet og bygget på Danmark
Tekniske Universitet. Dette stod i kontrast til test af metaller en der en standardiseret test på
højhastighedsmaskinen og hvor konstant tøjningshastighed kan opnås. Konklusionen blev at de
testmetoder der er succesfulde for metaller ikke var gældende for fiberforstærket materialer, på
grund af deres elastiske opførsel og lave brudtøjning. Dette er problematisk da
højhastighedsmaskinen kunne teste ved tøjningshastigheder hvor andre metoder var praktisk
umulige.
Ved højere tøjningshastigheder anvendes der en Split Hopkinson Pressure Bar. Denne metode blev
analyseret ved formulering af en ny bølgemekanik model for test emnet. Modellen muliggjorde
analyse af ligevægtsprocessen op til den homogene spændingstilstand uden antagelser om en
homogen spændingstilstand eller en konstant tøjningshastighed. Yderligere kunne modellen
håndtere stødstænger af forskellige materialer og diametre og indeholdte derfor ikke de
begrænsning som de modeller der eksisterede i litteraturen. Modellen viste at hvorvidt om den
konstant tøjningshastighed eller den homogene spændingstilstand indtræffer først, afhænger af
impedans forholdet mellem testemnet og stødstængerne. Det blev også vist at testemnet længde
ikke er styrende for den maksimalt mulige tøjningshastighed, men kun styre tiden det tager at opnå
den. Modellen blev sammenlignet med en række test udført på en Split Hopkinson Pressure Bar
ved University of Southampton. Litteraturen har vist at test emnet i en Split Hopkinson Pressure
Bar skal belasted med en konstant acceleration for at opnå en homogen spændingstilstand og
konstant tøjningshastighed før brud. I dette arbejde blev det vist at en bredt anvendt ”pulse
shaping” metode til at genere den konstant acceleration var begrænset i hvor høj en acceleration
der kunne opnås og derved var metoden begrænset. Det blev dog konkluderet at så længe
acceleration kunne styres tilfredsstillende så var det muligt at opnår en konstant tøjningshastighed
og homogen spændingstilstand før brud.
Department of Mechanical Engineering
VII
Department of Mechanical Engineering
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1 Introduction
The stress-strain behaviour of fibre reinforced polymeric (FRP) composites materials, can be
highly strain rate dependent. Typically, the stiffness and strength increases with increasing applied
strain rate and the overall behaviour becomes more brittle. FRP materials are becoming more
widely used in primary and secondary structures in engineering applications, where weight is a
critical factor. These applications include both military and civilian structures such as vehicles and
airplanes, and often the applications are exposed to dynamic and transient impact that causes
damage at high strain rates. Some parts of the structure are exposed to very high strain rates,
whereas others are subjected to lower strain rates. Simulation of the events incorporates materials
models that describe the material behaviour at different strain rates and they are calibrated from
material tests performed at elevated strain rates.
The established high strain rate test methods are aimed at metallic materials[1], however, no
dedicated high strain test methods and procedures exist for fibre reinforced polymeric composite
materials. The field remains a research task, which adapts methods from the metallic materials
testing to testing of FRP materials.
The anisotropic behaviour of FRP materials makes it a cumbersome and costly task to measure all
material parameters at elevated strain rates. Strain rate effects can be divided into a viscous elastic
effect and a strength effect. The viscous elastic effects are a change of the elastic response with
strain rate, and, it is not given that all constitutive parameters has the same strain rate sensitivity.
The strength effect is a change in failure stress and failure strain with strain rate and it is not either
given that the failure properties possess the same strain rate sensitivity. The experimental
procedures are often tailored to the specific material parameter(s) to be estimated [2] and further
specialisation of the equipment is needed when the strain rate is increased. The test set-ups become
highly complex and factors such as inertia come into play, which complicates the data
interpretation and validation. Figure 1.1 shows a classic overview of strain rates, prevailing test
methods, and considerations. At strain rates up to 0.1 /s, the range covered by standard test
machines, the process is considered Isothermal, and inertia forces can be ignored. Above 0.1/s,
inertial force must be taken into consideration in the experimental set-up and the test equipment
becomes specialised. Heavily modified servo-hydraulic test machines are used to access the strain
rate range. Further, the deformation process becomes adiabatic and the test specimens may
experience severe heating. The Split Hopkinson Pressure Bar (SHPB) is the preferred method from
200/s and up to 104 /s dependent on the test material. If higher strain rates are to be achieved, the
specimen must be brought into a unidirectional strain state, instead of the unidirectional stress state
normally used at lower strain rates[3].
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Figure 1.1 Dynamics aspects of mechanical testing [4]
A list of test methods sorted by test type is given in Table 1.1. Reviews of FRP material testing at
high strain rate showed that the Split Hopkinson Pressure Bar and the high-speed servo-hydraulic
test machine were the preferred methods for testing FRP materials [4-6]. The combination of these
methods covers a strain rate regime from 0.1 all the way up to more than 1000 /s. Further, these
are also the standardised methods for testing metallic materials [7-9]. The car industry particularly
has driven the standardisation of high strain rate testing of metallic materials in the high-speed
servo-hydraulic test machine [7, 10-13].
Table 1.1 Strain rates and different testing techniques adopted from [14]
Applicable strain rate, /s Testing Technique
Compression tests
<0.1 Conventional load frames
0.1 - 500 High speed servo hydraulic test machine
0.1 - 500 Cam plastometer
200 – 104 Split Hopkinson Pressure Bar
103 – 105 Taylor impact test
Tension test
<0.1 Conventional load frames
0.1 - 500 High speed servo hydraulic test machine
200 – 5x103 Split Hopkinson Tension Bar
104 Expanding ring
>105 Flyer plate
Shear and multi axial tests
<0.1 Conventional load frames
0.1 - 500 High speed servo hydraulic test machine
10 – 103 Torsional impact
100 – 104 Split Hopkinson Pressure Bar (Shear/Torsion)
103 – 104 Double notch shear and punch
104 – 107 Pressure-shear plate impact
It is reasonable to take already established methods and transfer them to other materials such as
FRP. However, in high strain rate testing, there is a great interaction between the test specimen
and the test machine and method, which may work for one type of material, but may not work for
another. In this work, only advanced FRP materials with highly aligned fibres and fibre volume
content above 50 % are considered. These types of material behave in many cases elastic up to
Department of Mechanical Engineering
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fracture, and do not exhibit plasticity before fracture as with metallic materials. Further, the
absorbed deformation before failure is very slight compared to metallic materials, due to low
failure strains. Depending on the lay-up of the fibres, FRP materials are also highly orthotropic,
which complicates the specimen design compared to the isotropic metallic materials. The
differences entail that the standards for metallic materials cannot be used as direct guidelines for
designed high strain rate testing of FRP materials and modification has to be applied. Hamouda et
al. [14] listed the important issues for dynamic testing of FRP materials:
1. Devising launch mechanism to produce the desired deformation state
2. Fixing the specimen in the test set-up
3. Selection of specimen geometry
4. Test duration and equilibrium time
5. Measuring transient parameters accurately
6. Data collection, management, and interpretation.
The launch mechanism may need a different set-up to produce the desired deformation state, since
FRP materials react differently than metallic materials. Fixing the specimen is also more
complicated due the orthotropic behaviour of FRP materials and the weak polymers around the
fibres. This is also connected to the specimen design, which is significantly altered compared to
metallic materials. Stress and strain are normally measured as average quantities over the
specimen, and that implies that the specimen is assumed to be in a homogenous stress state. In a
high strain rate test there may not be time to reach a homogenous stress state and this is more
pronounced for FRP materials due to low failure strains and low wave velocities compared to
metallic materials. The measurement itself of stress and strain on the specimen is complicated and
inertia forces may corrupt the recorded signals and this depends on how close the sensors can be
put to the specimen. The data interpretation may be different for FRP materials as they response
differently to deformation.
This work has focused on establishment of robust experimental test methods with the high-speed
servo-hydraulic test machine and the SHPB test rig. The high-speed servo-hydraulic test machine
was used for tensile testing whereas the SHPB was used in compression.
1.1 Thesis structure and novelty
The next sections of this chapter describe the concepts of strain and strain as used throughout the
work. Then the basic concept of dynamic deformation and wave mechanics is introduced, as well
as a brief overview of results from literature of high strain rate testing of FRP materials.
Chapter 2 describes the high-speed servo-hydraulic test machine in depth. Then, to
examine the interaction between the test specimen, the machine, and the limitations of
the method, a new model of the load train of the machine is presented.
Chapter 3 describes the design and construction of a high-speed servo-hydraulic
machine, which was done as a part of this work.
Chapter 4 describe a test series carried out at the constructed test machine. Parallels are
drawn to the work in chapter 3.
Chapter 5 describes the Split Hopkinson Pressure bars. A new method for calibration of
the bars is proposed in this chapter. Further, a new examination of the widely used pulse
shaping technique is given together with its limitations.
Chapter 6 describes a new generalised analytical wave mechanics model of the
specimen response for linear elastic materials in the SHPB. The model was used to set
up a design algorithm for specimen design, and to assess the limitation in achievable
strain rates.
Chapter 7 presents a new momentum trap method for the SHPB test rig. The method
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was easier to implement in an existing SHPB test rig than known methods from the
literature.
Chapter 8 presents a set of tests on an SHPB test rig performed with the methods
described in chapters 5, 6 and 7.
Chapter 9 summarise the work and draws a perspective.
Appendix A is a description of high-speed imaging. High-speed imaging has been
employed throughout the work together with Digital Image Correlation where possible
for verification of the measurement.
References to the remaining appendixes are given throughout the thesis.
The sequence of the chapters does not represent the chronological order of the work. The work in
Chapter 2 was first completed after chapters 3 and 4, as this work gave the necessary experience to
finalize the model in chapter 2. The work in chapter 6, especially that dealing with specimen
design, was first fully completed after the test in chapter 8 was completed.
The work in chapters 7 and 8 was carried out on an SHPB test rig at the University of
Southampton, whereas the work in chapters 5 and 6 was partly carried out at the University of
Southampton. The remaining work was carried out at the Technical University of Denmark.
1.2 Strain
Strain is the relative measure of deformation. An example is a bar with original length L deformed
by dL to final length Lf as in Figure 1.2. The common methods for calculating strain from this
deformation are listed in Table 1.2 along with their equations.
Figure 1.2 Bar deformed from initial length L to final
length Lf with the increment dL
Table 1.2 Formulas for average strain measures after deformation dL
Engineering strain Stretch ratio True strain Green strain Almansi strain
𝜀𝑡𝑒𝑐ℎ =𝑑𝐿
𝐿 𝜆 =
𝐿𝑓
𝐿 𝜀𝑇𝑟𝑢𝑒 = 𝑙𝑛 (
𝐿𝑓
𝐿) 𝜀𝐺𝑟𝑒𝑒𝑛 = (𝜆2 − 1) 𝜀𝐴𝑙𝑚𝑎𝑛𝑠𝑖 =
1
2(1 −
1
𝜆2)
Using the engineering strain as a reference, the differences in strain measures were examined. L
was set to 1 and Lf was varied between 0 and 0.05. The different strain measures were calculated
and plotted in Figure 1.2A. The relative deviations of the strain measures were calculated and
plotted in Figure 1.2B as a function of dL.
f
d
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A) Strain measure as function of the deformation dL
for a bar of length L=1.
B) Relative deviation of strain compared to the
engineering strain measure.
Figure 1.3 Strain measures and relative deviation compared to the engineering strain measure.
1.3 Strain rate
Strain rate is defined as the rate of change of strain with respect to time written as
𝜀̇ =𝑑𝜀
𝑑𝑡 (1.1)
If dL is applied during a time period dt, the deformation velocity is 𝑉0 =𝑑𝐿
𝑑𝑡. It follows from the
definition of engineering strain and equation (1.1) that the engineering strain rate is
𝜀�̇�𝑛𝑔 =𝑉0
𝐿 (1.2)
In terms of engineering strain, the strain rate is proportional to the deformation velocity. The
differences in strain rate measures were examined by applying the deformation dL in Figure 1.2
with a constant velocity V such that the engineering strain rate equalled 1. The corresponding
strain rates for the other strain measures were calculated from equation (1.1) and Figure 1.4A
shows the result. The engineering strain rate was used as reference, and the relative deviation was
calculated. Figure 1.4B shows that within an applied deformation of 5% engineering strain, the
engineering and true strain rates will deviate less than 5%. FRP materials deform up to 5-7%
technical strain [15] but will likely fail at less strain, especially for UD materials in the fibre
direction. Thus, the engineering strain can be used as a measure for strain and strain rate without
deviating substantially from the true strain and strain rate. The engineering strain and strain
measures are used throughout this work.
0 0.01 0.02 0.03 0.04 0.050
0.01
0.02
0.03
0.04
0.05
Deformation
Stra
in
Engineering strain
True strain
Green strain
Almansi strain
0 0.01 0.02 0.03 0.04 0.05-10
-5
0
5
10
Deformation
Dif
fere
nce
to
en
gin
eeri
ng
stra
in (%
)
Engineering strain
True strain
Green strain
Almansi strain
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A) Strain rate as function of the applied deformation B) Deviation of strain rate compared to the
engineering strain rate
Figure 1.4 Calculated strain rate as function of the applied deformation
1.4 Dynamic deformation
Equation (1.2) assumes implicitly equilibrium conditions over the specimen as the deformation per
time is averaged over the length L. When a dynamic approach is used instead, the strain rate
depends on how the deformation velocity is applied. The concept of stress waves in solids is used
to establish a closer look at the development of deformation in an elastic solid. 1D stress wave
motion in a slender elastic homogenous bar is described by the wave equation [3, 16]
𝜕2𝑢
𝜕𝑡2= 𝐶0
2𝜕2𝑢
𝜕𝑥2 (1.3)
C0 is the 1D wave propagation velocity calculated as
𝐶0 = √𝐸
𝜌 (1.4)
E is the elastic modulus along the propagation direction and ρ is the density of the bar material.
D’Alembert’s solution applies [3, 16] so the displacement u, velocity V, acceleration a and stress σ
-measures are given by
𝑢(𝑥, 𝑡) = 𝐹(𝑥 − 𝐶0𝑡) + 𝐺(𝑥 + 𝐶0𝑡)
𝑉(𝑥, 𝑡) =𝜕𝑢(𝑥, 𝑡)
𝜕𝑡= 𝐶0(−𝐹′(𝑥 − 𝐶0𝑡) + 𝐺′(𝑥 + 𝐶0𝑡))
𝑎(𝑥, 𝑡) =𝜕2𝑢(𝑥, 𝑡)
𝜕𝑡2= 𝐶0
2(𝐹′′(𝑥 − 𝐶0𝑡) + 𝐺′′(𝑥 + 𝐶0𝑡))
𝜎(𝑥, 𝑡) =𝜕𝑢(𝑥, 𝑡)
𝜕𝑥= 𝐸(𝐹′(𝑥 − 𝐶0𝑡) + 𝐺′(𝑥 + 𝐶0𝑡))
(1.5)
F and G are non-harmonic functions that describe the wave shape. F describes a wave that travels
in the positive defined direction of the bar whereas G describes a wave traveling in the negative
direction. If the deformation dL in Figure 1.2 is applied as an impact with a velocity V0, the
deformation will propagate through the bar at the elastic wave velocity as illustrated in Figure 1.5,
and the stress associated with the deformation propagating is [3]
0 0.01 0.02 0.03 0.04 0.050.5
1
1.5
Deformation
_"(=s)
Engineering strain rate
True strain rate
Green strain rate
Almansi strain rate
0 0.01 0.02 0.03 0.04 0.05 -10
-5
0
5
10
Deformation
Rel
ativ
e d
iffe
ren
ce (
%)
Engineering strain rate True strain rate Green strain rate Almansi strain rate
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𝜎 = 𝜌𝐶0𝑉0 (1.6)
Figure 1.5 Deformation of bar – distribution of wave deformation
At time a arbitrary time t1>t0 the velocity change is distributed over the distance dx = C0·t1 into the
material and the deformation simultaneously reaches a value of dL = V0·t1. The deformation has
then been distributed over a distance dx and the technical strain in this region is
𝜀0 =𝑑𝐿
𝑑𝑥=
𝑉0
𝐶0
(1.7)
The strain in equation (1.7) can also be found directly from equation (1.6) using Hook’s law
𝜀0 =𝜌𝐶0𝑉0
𝐸=
𝑉0
𝐶0
(1.8)
Thus, in the event of an instantaneous acceleration of one end of the bar to a constant velocity V0,
the velocity will distribute through the bar at a finite velocity C0 and leave a strain field with
constant amplitude behind its front. V0 is called the particle velocity. The strain rate is very high at
the passages of the wave front through the un-deformed material, but elsewhere the strain rate is
zero. The wave interacts with the interface at the clamped end. An interface is any change in the
total impedance where impedance is defined as [16]
𝑍 = 𝜌𝐶 = 𝜌√𝐸
𝜌= √𝜌𝐸 (1.9)
The total impedance is defined as
𝑍𝑇 = 𝜌𝐶𝐴 = √𝜌𝐸𝐴 (1.10)
A is the cross sectional area of the bar. To describe the interaction with the interface, basic
reflection theory is introduced. A stress wave in an elastic solid will reflect and transmit across an
interface. Three waves are defined:
𝜎𝐼 is the Incident wave
𝜎𝑅 is the Reflected wave
𝜎𝑇 is the Transmitted wave
Figure 1.5 shows the interface with waves defined.
t
t t
d t d t
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Figure 1.6 Interface between two slender bars
If no material is superimposed or gaps are created, the interface must be in equilibrium in terms of
forces and particle velocities. From this the basic reflection and transmission equation is derived.
The transmission and reflection coefficients can be calculated as [3, 16]
𝜎𝑇
𝜎𝐼
=2𝐴1𝜌2𝐶2
𝐴1𝜌1𝐶1 + 𝐴2𝜌2𝐶2
(1.11)
𝜎𝑅
𝜎𝐼
=𝐴2𝜌2𝐶2 − 𝐴1𝜌1𝐶1
𝐴2𝜌2𝐶2 + 𝐴1𝜌1𝐶1
(1.12)
When the wave reaches the clamped end, the wave reflects and transmits. If the clamped end is
completely rigid (E = ∞) such that complete reflection occurs the strain (and stress) doubles to 2 εo
in the reflection and propagates back to the other end. This is seen by rewriting equation (1.12) as
𝜎𝑅
𝜎𝐼
=1 + 𝐴1𝜌1𝐶1/𝐴2𝜌2𝐶2
1 + 𝐴1𝜌1𝐶1/𝐴2𝜌2𝐶2
≅ 1 ⇒ 𝜎𝑅 = 𝜎𝐼 (1.13)
Using equation (1.5), the stress sum of two waves that travel in each direction but overlap each
other becomes
𝜎 = 𝜎𝑅 + 𝜎𝐼 = 2𝜎𝐼 (1.14)
At a materials point the strain rate will be very high when the wave front passes, but zero at all
times during the deformation. Therefore, equation (1.2) is not representative for the strain rate
destribution in the bar, as only the average strain rate is given. The equilibrium process is
visualised in Figure 1.7 with a LS Dyna simulation of a slender bar with rigid boundaries at both
ends. The plot shows strain at different points in the bar as function of time. The strain rises
quickly when the wave passes, and then comes to a stable level before the wave passes again. The
shown material point (elements) only feels a high strain rate when the wave passes. During the
remaining time, the strain rate is zero, or close to zero. Furthermore, when both ends of the bar are
connected to rigid barriers, stress equilibrium will never be acquired if stress equilibrium is taken
as no stress difference between the ends of the bar. The stress difference is always the stress given
by equation (1.6). However, if the definition of stress equilibrium is relaxed, such that the stress
equilibrium is calculated as the stress difference divided by the average stress, a “dynamic”
equilibrium can be reached. The relative stress difference is calculated as in equation (1.15) with
Sa and Sb representing the stresses at the end of the bar.
𝑅 = 2 |𝑠𝑎 − 𝑠𝑏
𝑠𝑎 + 𝑠𝑏
| (1.15)
The equilibrium is well established within the split Hopkinson Pressure Bar (SHPB) community
and commonly used with R = 0.05 as the acceptance criteria for stress equilibrium [17-22].
I
T
Interface
A A
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Figure 1.7 Simulation with LS Dyna of a 0.2m long
and 0.02m wide bar. The nodes are rigidly clamped in
one end and in the other end the nodes are given an
initial velocity of 20m/s which is maintained through
how the entire simulation. E = 1GPa, ρ=1000kg/m3.
Poisson’s ratio is set to zero to remove dispersion
noise from the plot.
Equation (1.6) gives some interesting observations about the maximum impact velocity for a given
material. Shifting σ to σtu, the ultimate tensile strength for linear elastic materials the maximum
velocity becomes
𝑉0𝑚𝑎𝑥 =𝜎𝑡𝑢
𝜌𝐶 (1.16)
At V0 max the material fails instantaneously upon loading at the loaded end. Equation (1.16) is also
appropriate for materials with a yield point if σtu is changed to the yield stress σy. Then the
material is predicted to yield instantaneously [3]. Table 1.3 compares the maximum V0 max for steel
and aluminium with the same yield strength. The steel is penalised by its higher density and
requires three times higher yield strength than the aluminium to handle the same V0 max as
aluminium. The yield strength has to scale with the density to maintain the maximum impact
velocity as seen in equation (1.16).
Table 1.3 Maximum impact velocity V0 max for steel and aluminium with the same yield strength
Steel Aluminium
Yield strength 500MPa 500MPa
Wave velocity 5000 m/s 5000 m/s
Density 7900 kg/m3 2800 kg/m3
V0 max 13 m/s 36 m/s
For FRP materials, the stiffness and density depends on the fibre type(s), the lay-up, and the
matrix. Figure 1.8 gives an overview of impact velocities for different material systems listed in
the CSE Edupack material database [23]. The database gives a range of values for each material
parameter for each material and height of the bars reflect the range of the parameters going into
equation (1.16) which is used for Figure 1.8.
0 100 200 300 400 5000
2
4
6
8
10
Time (s)
Stra
in (%
)
Moving end
Middle
Clamped end
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Figure 1.8 Maximum impact velocity for FRP material systems. X-axis is material systems.
This maximum impact velocity is only an upper limit at which the material will fracture
immediately on impact. It does not help to explain if homogenous stress state can be reached
before the specimen fractures at impact velocities lower than the maximum velocity. In practice,
the impact velocity is almost never instantaneously applied; a gradual change of velocity is always
applied. One key point in high strain rate testing is to control the acceleration to the test specimen
so that a “dynamic” homogenous stress state is reached before specimen failure. Further, the
velocity should also be applied in order that the average strain rate approaches a constant level
before specimen fracture. This is a major concern of both chapters 2 and 5, which describe the two
test methods considered in this work.
1.5 Strain rate sensitivity of FRP materials
FRP materials have multiple factors that influence the strain rate sensitivity. The factors include
the fibre type, the fibre lay-up, the volume fraction, and the matrix type. It is difficult to compare
the data in the literature since these factors vary from paper to paper. Instead, to give an overview,
the methods from Barre et al. and Jacob et al. [5, 6] are adopted, where the effect of strain rate is
denoted as “Increase”, “Decrease” and “No effect”. Only results for F P materials obtained with
the split Hopkinson pressure bar and a high servo-hydraulic test machine were considered. The
emphasis was put on the elastic modulus E, the ultimate failure stress and failure strain. The data
are presented in Table 1.4.
Maxim
um
im
pact
velo
city
(m
/s)
0
20
40
60
80
100
120
140
160
180
200
Polyester (glass fiber, preformed, chopped roving)
Phenolic/E-glass fiber, woven fabric composite, quasi-isotropic laminate
Epoxy SMC (glass fiber)
Epoxy/HS carbon fiber, woven fabric composite, biaxial lamina
Phenolic/E-glass fiber, woven fabric composite, biaxial lamina
Polyester SMC (20% glass fiber, self-extinguishing)
Polyester (glass fiber, preformed, chopped glass)
Epoxy SMC (carbon fiber)
Glass/epoxy unidirectional composite
Epoxy/HS carbon fiber, UD composite, 0° lamina
PEEK/IM carbon fiber, UD composite, 0° lamina
Epoxy/S-glass fiber, UD composite, 0° lamina
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Table 1.4 Collected material data from literature
Abbreviations
SHPB Compressive Split Hopkinson tests
SHTB Tensile Split Hopkinson tests
VHS High-speed servo-hydraulic test machine. (Static properties was obtained with standard methods)
PP Polypropylene
PE Polyethylene
Pol Polyester Vin Vinyl ester
PPTA Polyphenylene terephthalamide
HMPE High Modulus Polyethylene fibre
E Elastic modulus
𝜎𝑢 Ultimate failure stress
𝜀𝑢 Ultimate failure strain
Material Test Strain rates /s Results Referenc
e
Crossply Carbon /Epoxy SHPB Static – 817
Increase in E and
𝜎𝑢
Decrease in 𝜀𝑢
[24]
UD 0° & 90° Carbon /Epoxy SHPB Static – 817
Small increase in
E and 𝜎𝑢 Small decrease in
𝜀𝑢
[24]
UD 0° Carbon/Epoxy SHTB Static - 450
No effect on E
and 𝜎𝑢
[25]
Quasi Isotropic Carbon /Epoxy SHTB Static - 145 Increase in 𝜎𝑢
Decrease in 𝜀𝑢 [26]
Satin Weave Carbon/Pol SHPB Static-1000
Increase in E and
𝜎𝑢
No effect on 𝜀𝑢
[27]
Plain Weave Carbon/epoxy SHTP Static – 2000 𝜎𝑢 Unchanged
Ref.
16&17 in [6]
UD Carbon/Epoxy VHS Static - 45 Increase in 𝜎𝑢 [28]
Carbon Fibres SHTP Static – 1300 E, 𝜎𝑢 and 𝜀𝑢 Unchanged
[29]
3D braided Glass/ epoxy SHTP Static - 2800
Increase in E and
𝜎𝑢
Decrease in 𝜀𝑢
[30]
Plain weave - Aramid/PP SHTP Static - 1000 Increase in 𝜎𝑢
Decrease in 𝜀𝑢 [31]
Plain weave - Aramid/PET1 SHTP Static - 1000 Increase in 𝜎𝑢
Decrease in 𝜀𝑢 [31]
Plain weave - Aramid/PET2 SHTP Static - 1000 Increase in 𝜎𝑢
Decrease in 𝜀𝑢 [31]
Plain weave - PE/PE SHTP Static - 1000 Increase in 𝜎𝑢
Decrease in 𝜀𝑢 [31]
UD - PE/PE SHTP Static - 1000 Increase in 𝜎𝑢
Decrease in 𝜀𝑢 [31]
Satin Weave Aramid/Pol SHPB Static-1000
Increase in E and
𝜎𝑢
No effect on 𝜀𝑢
[27]
PPTA fibres a and b (Aramid) SHTP Static- 850
Decrease in E and
increase in 𝜎𝑢
Increase in 𝜀𝑢
[32]
PPTA fibres (Aramid) SHTP Static - 850
Decrease in E and
Decrease in 𝜎𝑢
Decrease in 𝜀𝑢
[32]
HMPE Fibres SHTP Static– 800
Decrease in E and
increase in 𝜎𝑢
Increase in 𝜀𝑢
[32]
HMPE fibres (Spectra900) SHTP Static -300
Decrease in E and
increase in 𝜎𝑢
Decrease in 𝜀𝑢
[33]
Spectra1000 laminate shield SHTP Static-850
Increase in 𝜎𝑢
below 400/s Increase above
400/s
[34]
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PPTA fibres (Aramid) SHTP Static – 400
Increase in E and
increase in 𝜎𝑢
Increase in 𝜀𝑢
[35]
Plain/ Satin weave /UD Glass/Pol SHTP Static – 2000 Increase in 𝜎𝑢
Ref. 16&17 in
[6]
Plain/ Satin weave /UD
Glass/Epoxy SHTP Static – 2000 Increase in 𝜎𝑢
Ref. 16&17 in
[6]
Plain weave Glass/Epoxy SHPB
Static -1000
Increase in E and
𝜎𝑢
[36]
UD Glass/Epoxy VHS Static - 500
Increase in E and
decrease in 𝜎𝑢
and 𝜀𝑢
[37]
UD Glass/Epoxy VHS Static -100 Increase in 𝜎𝑢 [28]
UD Glass/Pol VHS Static -100 Increase in E and
𝜎𝑢 [38]
Plain Weave Glass/Epoxy SHTP Static -400 Increase in 𝜎𝑢 [39]
Commingled plain weave E-glass/polypropylene
Drop test
(Tension/
Comp)
Static - 120
Increase in E and
𝜎𝑢
Increase in 𝜀𝑢
[40]
S-2 Glass/Epoxy SC-15 SHPB
Quasi-1100
Increase in E and
𝜎𝑢 Slight decrease in
𝜀𝑢
[41]
Plain weave S2-Glass/Vin SHPB
Quasi - 800
Increase in E and
𝜎𝑢
Increase in 𝜀𝑢
[42]
Crossply Glass Epoxy VHS Quasi - 100
Slight increase in
E and 𝜎𝑢
Increase in 𝜀𝑢
[43]
The most commonly studied materials in the literature are the glass/epoxy (GFRP) and
carbon/epoxy (CFRP) in different configurations. Based on the data in Table 1.4 GFRP appears to
be more rate sensitive in all considered parameters than CFRP, a conclusion also reached by Barre
et al. [6]. In addition, Barre et al. concluded that weaved fibre structures for GFRP were more rate
sensitive than UD. Furthermore, plain weave was more sensitive than a five satin weave. This
indicates that the rate sensitivities are related to the elastic interactions between the matrix and
fibre, since a satin weave has less elastic interaction than a plain weave. Zhou et al. [29] presented
a comparison of fibre bundles and found the carbon fibres were insensitive to strain rate whereas
the glass fibres were highly sensitive as shown in Figure 1.9.
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Figure 1.9 Comparison of fibre bundle tensile strength (T-700
= Carbon fibre) [29]
The specific type of fibre and Matrix can have a great influence on the behaviour as shown in
Figure 1.10.
A) Rate sensitivity of tensile strength
B) Rate sensitivity of failure strength
Figure 1.10 Tensile strength and failure strain for Aramid and polyethylene fibres (SK66 and UD66 ) [31]
The brief overview of data given here indicates that many factors affect the rate sensitivity of FRP
materials. An examination of Table 1.4 shows that contradictions exist in some of the obtained
results in the literature, a conclusion also reached by others [4-6]. For example, do PPTA fibres
and UD CRFP materials have the ability to both increase and decrease in strength? These
contradictions highlight the importance of examining the test methods to identify potential
improvements and to understand their limitations.
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2 Examination of the servo-hydraulic test-machine as a high speed loading device
High-speed servo-hydraulic test machines are becoming widely used in the automotive industry to
test sheet metals at strain rates between 1-500/s [7], and a standardised test method was developed
based on years of research [9]. This work has spread to other types of materials and a test standard
was developed for pure polymers [44], but not to fibre-reinforced polymers. Several papers
reported the use of high-speed servo hydraulic machines [6, 38, 45-49] for testing FRP materials at
elevated strain rates. However, different specimen geometry, different gripping methods, and
different load measurements methods were used. The variety of methods demonstrated the lack of
a standard in the field and gives space for an examination of the high-speed servo hydraulic test
machine for FRP material testing at elevated strain rates.
This chapter describes the working method of the high-speed servo-hydraulic test machine. Then a
set of ordinary second order differential equations is used to model the “load train” for testing FRP
materials and conclusion is drawn about the influence of the load train to the measured data.
Comparisons are made to real tests for validating the models.
2.1 Description of the high servo hydraulic machine
A servo-hydraulic test-machine consists of a hydraulic cylinder mounted into a steel frame. The
test specimen is mounted to the frame, and the piston rod of the hydraulic cylinder. The specimen
is deformed in either tension or compression by moving the piston rod. Pressurised oil is used as
medium to transfer kinetic energy to the piston rod. The oil is pressurised by a hydraulic pump and
a servo valve is used to control the oil flow to- and from the hydraulic cylinder and controlling the
deformation of the specimen. Figure 2.1 shows a schematic overview of a high-speed hydraulic
test machine.
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Figure 2.1 Schematic overview of the servo hydraulic test machine [1]. The addition of a
“Lost motion unit” in the load train is a major different to standard one axis hydraulic
test machines. The lost motion unit allows the piston rod to accelerate to a given velocity
before the specimen is loaded.
The maximum oil flow rate of the hydraulic system determines the maximum piston rod velocity.
Big hydraulic pumps delivers a flow rate of hundreds of litres per minute, but for velocities above
1 – 5 m/s, several thousand litres per minute is required dependent of the machine setup. Only a
small volume of oil is required at the high flow rates and normally hydraulic accumulators are
used for supplying the oil at these high flow rates. A hydraulic bladder accumulator consists of a
combined oil/gas cylinder with a bladder inside which separate the oil from the gas. The gas is pre-
set to a given pressure and during operation, the oil will flow into the accumulator until the gas is
compressed to the same pressure as the oil. The compression happens with a low oil flow rate, but
when the system pressure decreases the gas expand at rates up to its sonic speed and forces the oil
into the system.
The hydraulic cylinders are normally double acting with equal pressurised areas, and oil flow to
and from the cylinder is controlled with a (4/3)1 way servo valve. Oil flows to one side of the
cylinder while oil is allowed to escape from the other side, when the valve opens. Both the in- and
outgoing oil flow is going through the servo valve and the flow rate is controlled by the valve
opening and the pressure drop over the valve. As the oil start to flow into the cylinder the oil
pressure drops and the gas in the accumulator starts to expand and forcing oil into the system. As
the oil is flowing through the valve, a pressure drop occurs over the valve due to the flow
resistance that increases with flow rate. Valves are normally rated to a given flow for a given
pressure drop at a given opening of the valve. Flows for other pressure drops are calculated with
the square root function for sharp edged orifices [50]:
𝑄
𝑄𝑁
= √𝑃
𝑃𝑁
(2.1)
QN and PN are the rated flow and rated pressure drop and Q is an arbitrary flow with its associated
pressure drop P. When the oil flows out of the cylinder, it goes through the valve and directly to
tank on the pump. The tank is depressurised so there is no pressure on the outlet side of the valve
and a pressure must be build up to force the oil fast enough through the valve. Then, the pressure
1 (4/3) means four ports and three positions. The four ports are Pressure, Return, Channel A, and Channel B. In first
position Pressure is connected to Channel A, and Return to Channel B. In second position the Pressure is connected to Channel B and Return connected to Channel A. The last position is no connections.
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differential over the piston rod is the gas pressure in the accumulators, minus the pressure loss in
the ingoing side of valve, and minus the pressure build-up before the outgoing side of the valve.
Other hydraulic resistance and internal mechanical friction in the cylinder also reduces the
pressure differential over the piston rod. However, the main loss is over the valve for a high
quality system. This pressure differential is driving the acceleration of the piston rod and/or the
force, which deform the test specimen in the test machine. The piston rod velocity is first constant
when the pressure differential is zero and all pressure is lost in the valve and friction. Figure 2.2
shows the pressure losses when the valve opens, and when the piston rod has obtained constant
velocity.
Figure 2.2 Pressure drop at valve during opening and at steady piston
rod velocity. Pmax/2 is lost when oil passes the valve into the cylinder and
a pressure of Pmax/2 is build up in the cylinder as the oil has to pass out of
the valve. �̈� is the piston rod acceleration and �̇� is the piston rod velocity.
Commercial high-speed hydraulic test-machines reach 20 – 25m/s in piston rod velocity, utilizing
this principle of accumulator driven oil flow. Since the pressure differential is zero at constant
velocity, there is no force to deform the specimen upon impact with the specimen. Deformation is
only achieved through the kinetic energy of the piston rod and by deacceleration of the piston rod.
However, a pressure differential builds up as soon the the piston rod deaccelerates.
Hydraulic test-machines normally operate with oil pressures ranging from 210 – 280 bars where
high-speed servo-hydraulic machines tend to utilize 280 bars pressure. The machine can be
designed with smaller moving components at this pressure, while maintaining the performance
ig valve
ig valve
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comparing to using 210 bar oil pressure.
Standard test machines operate with a closed loop controller, which uses a feedback signal from
the test machine to adjust the piston rod movement. For high-speed servo-hydraulic test machines,
the closed loop control principle cannot be used, since the electrical and mechanical dynamics are
not sufficient at the high piston rod velocities [7]. Instead, open loop control is used where the
valve is opened and left at some constant opening until the test is over. There is no adjustment of
the opening during test and the system is unable to adapt to external forces, such as the impact of
the specimen.
2.1.1 Lost motion unit - Slack adapters
Material testing at quasi static rates is performed with piston rod velocities of 0.1-5mm/min
[51]with an acceleration length of few micrometres, whereas high-speed servo hydraulic test
machine requires 100 – 300 mm acceleration length to reach 20 – 30m/s (1.8e6 mm/min). This
acceleration length is far above the 1-3mm displacement to failure for a UD GFRP test specimen,
with a gage length of 50-150mm and a fracture strain of about 2%.
Devices as “slack adapters/lost motion units” allow the piston rod to accelerate before impact of
the specimen [1, 10, 12, 45, 47, 52]. The slack adapter converts the long acceleration length of the
piston rod to a short acceleration length for the specimen, thus aiming to accelerate the moving
end of the specimen to the test velocity before the specimen fractures. Figure 2.3 shows three
designs of slack adapters.
Figure 2.3 Different concepts for the slack adapter [12]
In (a) the piston rod accelerates to the desired velocity and impacts the lost motion rod (LMR).
Typically, a piece of soft material as rubber is put between the contact faces to avoid too high
contact forces. The thickness of the rubber layer will influence the acceleration length of the
specimen. The LMR is not rigidly connected to the piston rod and can bump between the impact
velocity and velocity higher than this [7, 13]. Approach (a) also adds extra mass to the specimen,
which causes inertial damping. Approach (b) is optimized with wedges to avoid rebounds but is
sensitive to matching geometries and alignment. Approach (c) is a preloaded unit, which moves
with the piston rod along an extended test specimen. The unit releases and grip the specimen when
the test velocity is reached. The benefit from this design is the absence of a lower grip, which
damp the accelerations of the specimen. This approach is widely used for testing sheet metals in
the car industry [7, 10, 53]. The specimen for approach (c) has a special design with an elongated
end, which is stronger than the gage area, and the required specimen design is shown in Figure 2.4,
along with straight test specimens used for FRP materials. The straight sided coupon has a very
long effective gage length in this configuration, which results in a low strain rate for high impact
velocities.
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A) specimen designs for metallic materials[7] B) Specimen designs for FRP. The special specimen
design for FRP materials is taken from [28].
Figure 2.4 Specimen design used with approach (c) in Figure 2.3.
Approach (c) in Figure 2.3 is developed of Instron ltd under the name “Fast Jaws” and this name
is used throughout this work.
2.1.2 Load measuring
Load can be measured with a strain gage based load cell, a piezo electric based loadcell, with a
dynamometer, a strain gage directly on the test specimen, or with a strain gage directly on the
grip(s). The following recommendation is reported in literature with respect to test of metallic
materials [1]:
1. Conventional strain gage instrumented load cell Strain rate < 10/s.
2. Piezoelectric load cell Strain rate < 100/s.
3. Strain gages mounted on specimen or static grip For all strain rates.
A rapidly changing load will cause the load cell to oscillate at its resonance frequency known as
ringing. For metallic materials, the oscillations are introduced at the yield point while for elastic
brittle materials they may appear from the beginning of the loading. Figure 2.5 shows this
behaviour together with qualitative guidelines for acceptance or rejection of load data with
oscillations.
A) Acceptance guidelines for
ringing in tests of metallic
materials [44]
B) Output from a piezo electric
loadcell for a test of steel at 10m/s
(150/s) [43]
C) Test of GFRP at 10m/s with
ringing in the load signal [28]
Figure 2.5 Examples of ringing and guideline for acceptance of ringing.
Some researches removes the oscillations by filtering of the data [28, 54], but filtering distorts the
underlying load response of the specimen, and instead,
1 1.2 1.4 1.6 1.8 20
5
10
15
20
25
30
35
Time (ms)
Load
(kN
)
Loadcell
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the oscillations should be minimized by other means before applying filtering [11]. Oscillations
happen to all load cells with a mass. The piezo electric load cells are in favour due to their high
resonance frequency. They can be used at high rates of loading without obscuring the load
measurement too much, but the performance depends on the grip mass attached to the load cell [9,
52, 55]. The resonance frequency of the load-measuring device can be maximized for metallic
materials by turning the test specimen into a load cell itself called a dynamometer. This
arrangement also benefits from the close position to the strain measuring as it reduces the phase
lag between the signals [1]. The dynamometer consists of a pair of strain gages mounted on an
area on the specimen, which deform with a known calibrated elastic behaviour during the entire
loading of the specimen. For metallic materials, the dynamometer is only calibrated in a load range
up to the yield load of the specimen. Load measuring above the yield load relies on extrapolation
of the calibration curve [1, 7]. Design recommendations for a dynamometer on dogbone specimens
for metallic materials are given by Wood et al. [53]. The dynamometer is difficult to implement on
FRP test specimens, since they are made as straight side specimens, which leaves no part of the
specimen to be undamaged all the way up to fracture.
2.2 Numerical model
The entire setup with load cell, grips, slack adapter, rubber layers etc. and the piston rod
constitutes the “load train”. For this work, the load train was viewed as a simple finite element
system consisting of a connected set of masses and springs. Approach (a) and (c) in Figure 2.3
constitute the two main types of slack adapters and they were modelled to examine the influence
of the mass distribution on the deformation rate of the test specimen. Figure 2.6 shows a schematic
overview of the two systems along with the terms of concentrated masses (nodes) and the forces.
Distributed masses were lumped to the concentrated masses.
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Figure 2.6 Schematic overview of slack adapters. Note the slack adapter is
running inside the piston rod for the conventional slack adapter.
The systems were modelled as one-dimensional dynamic system with the following assumptions:
A rubber layer was used to damp the impact between the two parts of the slack adapter
(Only for the Conventional slack adapter).
The piston rod was moving at constant velocity when the specimen was impacted.
The entire oil pressure was lost over the valve at constant velocity.
The piston rod had infinitely stiffness compared to the remaining parts of the load train.
The load cell was mounted rigidly to the frame of the test machine.
All masses were lumped.
Grips were infinitely stiff.
The specimen was rigidly connected to the grips.
The fast Jaws frame was rigidly connected to the piston rod and moved with the piston
rod as one unit.
The Fast Jaws frame was deformable.
Each mass node had a displacement x, a velocity �̇� and acceleration �̈�. Newton second law were
used to set up force equilibrium for each node and Table 2.1 gives the system of equations, which
describe the movement of the individual nodes.
m
-Flc
Fml
m
-Fml
Fs
m
-Fsl
Ffj
m
-Ffj
Foil
m
-Fs
Fsl
m
-Flc
Fs
m
-Fs
Fs
m
-Fs
Frub
m
-Frub
Foil
oadcell oadcell
pper grip
ower grip
Fast Jaws
Piston rod
Piston rod
Slack adapter
ubber layer
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Table 2.1 Mass distribution and equation systems for the slack adapter systems
Fast Jaws Conventional slack adapter
m1 = Mass of upper grip and load cell m2 = Mass of mini load cell and specimen
m3 = Mass of slack length of specimen
m4 = Mass of preloading unit of the Fast Jaws m5 = Mass of Fast Jaws frame and piston rod
m1 = Mass of upper grip and load cell m2 = Mass of lower grip
m3 = Mass of Slack adapter end
m4 = Mass of piston rod
𝑚1�̈�1 = 𝐹𝑚𝑙 − 𝐹𝑙𝑐 𝑚2�̈�2 = 𝐹𝑠 − 𝐹𝑚𝑙 𝑚3�̈�3 = 𝐹𝑠𝑙 − 𝐹𝑠 𝑚4�̈�4 = 𝐹𝑓𝑗 − 𝐹𝑠𝑙
𝑚5�̈�5 = 𝐹𝑜𝑖𝑙 − 𝐹𝑓𝑗
𝑚1�̈�1 = 𝐹𝑚𝑙 − 𝐹𝑙𝑐 𝑚2�̈�2 = 𝐹𝑠𝑙𝑎 − 𝐹𝑚𝑙 𝑚3�̈�3 = 𝐹𝑟𝑢𝑏 − 𝐹𝑠𝑙𝑎 𝑚4�̈�4 = 𝐹𝑜𝑖𝑙 − 𝐹𝑟𝑢𝑏
Flc = Force of the load cell Fml = Force of local dynamometer
Fs = Force of specimen
Fsl = Force of slack end of specimen
Fsla = Force of slack adapter
Ffj = Force of Fast Jaws frame
FRub = Force of Rubber damper Foil = Force on the piston rod from the hydraulic oil
Parameters
Klc = Stiffness of the load cell
As = Cross sectional area of the specimen
Es = Elastic modulus of the specimen Ls = Gage length of the specimen
Kml = Stiffness of the dynamometer
Ksl = Stiffness of the slack length of specimen Ksla = Stiffness of the slack adapter
Ksl = Stiffness of the Fast Jaws frame
σy = Yield stress εy = Yield strain
n = Hardening parameter
V = Piston rod velocity
Ap = Piston rod pressurised area
Q = Flow rate
P = Pressure drop QN = Rated flow rate
PN = Rated pressure drop
σR = nominal stress in rubber C = Neo Hookian law – material constant
λ = Stretch ratio
LR = Thickness of rubber damper AR = Cross sectional area of rubber damper
A Kistler 9071 Piezo electric load cell were considered as load cell which had a stiffness in the
loading direction of KLC = 26e9 N/m. The force in the load cell was calculated solely from the
displacement of node 1 as the load cell was rigidly connected to the test machine. The force
became
𝐹𝐿𝐶 = 𝐾𝐿𝐶𝑥1 (2.2)
Two types of specimens were considered: linear elastic, and an elasto-plastic material. The linear
elastic force response of both types of specimens were calculated with
𝐹𝑠 = 𝐾𝑠(𝑥2 − 𝑥1) (2.3)
Where Ks is defined as
𝐾𝑠 =𝐴𝑠𝐸𝑠
𝐿𝑠
(2.4)
The plastic force response of the elasto-plastic specimen were calculated by a rate independent
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power law
𝐹𝑠 = 𝐴𝑠𝜎𝑦 (𝜀
𝜀𝑦
)
𝑛
(2.5)
Equation (2.5) is only valid for a monotonic rising load. The non-reversible process in the
specimen, which occurs during an unloading phase, was represented with the algorithm in
Appendix D.
Fml,Fsl,Fsla and Ffj were formulated as the linear elastic response in equations (2.3) and (2.4). The
rubber damping were described with a one dimensional Neo Hookian law (Hyper elastic material)
formulated as
𝐹𝑅𝑢𝑏 = 𝐴𝑅𝐶 (𝜆 −1
𝜆2) (2.6)
The deformation measure for this law was the stretch ratio calculated as
𝜆 =𝑙
𝐿𝑅
(2.7)
l is the length after deformation and LR is the original length. In terms of node displacement for the
conventional slack adapter, equation (2.7) yield
𝜆 =𝐿𝑅 − (𝑥4 − 𝑥3)
𝐿𝑅
(2.8)
Since the slack adapter is running inside the piston rod as shown in Figure 2.6 the rubber becomes
compressed when (x4 – x3)>0. To create a compressive force in the rubber when (x4 – x3)>0, (x4 –
x3) were subtracted from the initial thickness instead of added as given in equation (2.8). The
rubber damper is not rigidly connected to the piston rod or the slack adapter so the rubber could
only exert compressive forces to the system. In the model this was imposed by setting FRub to 0
when (x4 – x3)<0. As explained later, the system of equation were solved with a explicit time
stepping algorithm and in some occasion the step forward in time could result in (x4 – x3)>LR
which corresponded to a final negative thickness. In this case the system were penalised by setting
λ in equation (2.6) to
𝜆 = 0.001 (2.9)
The force exerted to the system by the oil pressure over the piston rod was calculated from the
assumption of complete pressure loss in the actuator at constant velocity. The oil flow Q into the
cylinder was related to the piston rod velocity V by:
𝑉 =𝑄
𝐴𝑝
(2.10)
Inserting equation (2.10) into (2.1) yields
𝑃 =𝑃𝑁
𝑉𝑁2 𝑉2 (2.11)
The high-speed servo-hydraulic test machine was operated in open loop so the valve was opened
and kept open. The oil flow would then change until the pressure differential over the piston rod
was zero. When the specimen was impacted at constant velocity the piston rod was slowed down
and the flow rate decreased. The resulting net differential pressure over the piston rod then exerted
a force on the system. When the valve was set to a given opening, a constant velocity VN was
obtained with the pressure drop PN corresponding to the system pressure. The pressure drop at any
velocity V could be calculated from equation (2.11). If the velocity was Vx1, equation (2.11) gives
a pressure drop Px1 and if the velocity is Vx2 the associated pressure drop is Px2. Changing the
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velocity from Vx1 to Vx2 gives the following pressure drop over the valve
𝑑𝑃 = 𝑃𝑥1 − 𝑃𝑥2 (2.12)
This dP was the differential pressure over the piston rod, causing a force to the system as
𝑑𝐹𝑜𝑖𝑙 = 𝐴𝑝𝑑𝑃 (2.13)
Combining equations (2.11), (2.12) and (2.13) yields
𝑑𝐹𝑜𝑖𝑙 = 𝐴𝑝
𝑃𝑁
𝑉𝑁2 (𝑉𝑥1
2 − 𝑉𝑥22) (2.14)
In a simulation the impact velocity Vx1 would be the initial constant velocity of the piston rod
before the specimen impact such that
𝑉𝑥1 = 𝑉𝑁 (2.15)
Combining equations (2.14) and (2.15) and exchanging 𝑉𝑥2 to the piston rod velocity �̇�𝑖 yields
𝑑𝐹𝑜𝑖𝑙 = 𝐴𝑝
𝑃𝑁
𝑉𝑁2 (𝑉𝑁
2 − �̇�𝑖2) (2.16)
When �̇�𝑖 is negative, the piston rod moves in the opposite direction of the initial movement and oil
would be forced backward in the system. The full pressure will build up quickly, and and in this
case Foil is set to 𝐴𝑝𝑃𝑁.
The explicit ODE45 solver in MatLab was used to solve the differential equations for
displacement, velocities and accelerations.
2.3 Verification of models
The conventional slack adapter model was used for verification. A modified MTS high-speed
machine equipped with a conventional slack adapter was used for comparison. The test machine is
described in chapter 3.
First, a hypothetically case was simulated. Figure 2.7A shows the simulation where the slack
adapter impacts an infinitely strong linear elastic specimen at 20 m/s. The piston rod was given an
acceleration length of 230mm and a maximum velocity of 20m/s. First, the piston rod accelerated
and at point 1 the piston rod impact the slack adapter, and the slack adapter and lower grips moved
along with it. At point two, the specimen had stopped the piston rod, but the momentum of the
piston rod forced the specimen to a strain where the specimen exerted a larger force than the force
of the oil. The piston rod was pulled back and when the specimen reached zero strain at point 3 it
went to compression and stopped applying force to the piston rod. The piston rod continued due to
the non-rigid connection between the piston rod and slack adapter, and the force of the rubber
damper was set to zero. The piston rod was then deaccelerated by the oil pressure and bounces
back again. This process continued as damped oscillations until equilibrium between the force and
oil force was reached. The damping occurred due to the constant force of the oil when the velocity
of the piston rod was negative Figure 2.7B shows a comparison between a simulated acceleration
curve of the piston rod and a measured curve. The comparison showed good agreement between
the final velocities before impact of the slack adapter. However, the path to the final velocities
differed. The difference was accounted to the effect of internal friction in the cylinder and the
opening time of the valves. The valves used here was two Moog D665 with a valve opening time
of 10-15ms [50]. None of these two effects were taken into account.
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A) Simulation of conventional slack adapter. 1 is the
impact between the piston rod and the slack adapter.
At 2 the piston rod was brought to zero velocity by
the infinitely strong test specimen. At 3 the contact
between the specimen and piston rod is lost.
B) Comparison of simulated and measured velocity
of the piston rod.
Figure 2.7 Simulation of the conventional slack adapter.
The model was compared to a series of test of UD Eglass/Epoxy specimens tested at 7.5 and 20
m/s. The specimen parameters are given in Table 2.4 and Figure 2.8 shows the specimen design.
Appendix F gives the parameter values for the simulation, which was used for the comparison with
the tests.
Table 2.2 Specimen parameters for comparison tests
Parameter Description Value
Esp Elastic modulus of the specimen 42 GPa
Asp Specimen cross sectional area 15 mm2
Ls Specimen gage length 50mm
Figure 2.8 Schematic specimen design
A set of custom grips made from high strength aluminium was used for gripping the specimen .
Load was measured with a Kistler 9071 Piezo electric load cell.
Figure 2.9A shows a comparison between the test and the model for the strain vs. strainrate
response. In the simulation, the strain rate increased immediately, while in the real test the strain
rate raises slowly and then accelerate. Figure 2.9B shows the specimen strain plotted against the
piston rod position with a free travel of 185mm before impact.
0 20 40 60-100
-50
0
50
Time (ms)
Dis
pla
cem
ent
(mm
)
Upper grip
Lower grip
Slack adapter
Piston rod
1
2
3
0 5 10 15 200
5
10
15
20
25
Time (ms)
Vel
oci
ty (m
/s)
Simulation
CRBJ20D-01 LVDT
hspecimen
wspecimen/
wtab
tab
specimen
Tabs Tabs
S
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A) Specimen strain vs. strainrate B) Specimen strain versus piston rod position
Figure 2.9 Simulation and test of linear elastic material of straight side coupon specimen with cross section
=15mm2, Esp = 38GPa and Lg =50mm.
The plot reveals that the specimen strain up to 1% was built up only 15 - 25mm before impact with
a slowly exponential increase in strain rate. The strain may be a result of friction or trapped air
between the piston rod and slack adapter. Friction would most likely set of from the start of the
piston rod movement and would not build up towards the ending, unless the slack adapter became
progressively skew towards the impact end. Such defects on the slack adapter were not observed.
Trapped air could cause the observed effect. To check if the observed effect could be trapped air,
the model was enhanced to predict the pressure rise inside the slack adapter as the piston rod
moved. The model is presented in Appendix C. The model handles the compressibility of the air
and models the outflow of air together with the pressure increase in the slack adapter. Any size of
outlet holes/leakage could be handled. The model output a force as function of the piston rod
position to a given time and model was implemented in the slack adapter model by adding the
force to the rubber and the piston rod. Trapped air would act against these parts of the load train.
The air was able to escape between the moving parts in the comparison tests and the escape cross
sectional area was estimated to around 15mm2. The initial air pressure in the slack adapter was set
to one atm. Figure 2.10 shows a comparison between a simulation and a test at 7.5 m/s impact
velocity. The air inclusion created the effect of slowly rising strain rate and a following rapid
increase. However, the air compression model does not account fully for the effect, but the model
was still found to describe the behaviour of the slack adapter reasonable accurate.
0 0.5 1 1.5 2 2.5 3 3.50
50
100
150
200
Strain (%)
Stra
in r
ate
(/s)
Real test, 7.5m/s
Simulation
165 170 175 180 185 0
1
2
3
4
5
6
Piston rod position
(mm) St
rain
(%
)
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A) Strain rate vs strain with air inclusion simulated B) Specimen strain vs piston rod position
Figure 2.10 Comparison between simulation and real test with air model included. The pressure was added
as pressure on the rubber and on the piston rod.
2.4 Examination of stress and strain rate response
Simulations were carried out to examine how the stress-strain curves are affected by the dynamic
loading conditions and then how the strain rate developed in the specimen as function of the
deformation history. The stress in the specimen was calculated by dividing the force of the load
cell with the initial cross sectional area of the specimen. In a real test, the strain would be
measured directly on the specimen with a strain gage or non-contact measurement methods. In the
simulations, the strain was calculated by dividing the elongation of the specimen with the initial
gage length. The deformation applied to the specimen was calculated as the displacement between
the two nodes surrounding the specimen. Further, also the forces in the slack adapter were
examined as the high acceleration forces could lead to destructive loading of the slack adapter
system. The effect of the missing rigid connection in the slack adapter was also studied. Four
setups were considered:
1. Conventional slack adapter – without any damping
2. Conventional slack adapter – with 3 mm rubber damper
3. Conventional slack adapter – with 3 mm rubber damper and air damping (fully closed
volume)
4. Fast Jaws system.
A linear elastic specimen was considered and the specimen parameters are given in Table 2.3.
Other parameters are given in Appendix F. The specimen design for the conventional slack adapter
is shown in Figure 2.9 while the specimen design for the Fast Jaws system is shown in Figure
2.13. The model of the Fast Jaw setup were adjusted such that the specimen “slack end” was
accelerated to piston rod velocity in less than 15 µs, which corresponded to the acceleration time
reported in literature [12].
0 0.5 1 1.5 2 2.5 3 3.50
50
100
150
200
Strain (%)
Stra
inra
te (
/s)
Real test 7.5m/s
Simulation
0 0.05 0.1 0.150
0.5
1
1.5
2
2.5
3
3.5
Pistonrod position (m)
Stra
in (%
)
Real test 7.5m/s
Simulation
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Table 2.3 Specimen parameters and their nominal values
Parameter Parameter Elastic specimen Elasto plastic specimen Varied
Esp Elastic modulus 42 GPa 200 GPa x
Asp Specimen cross sectional area 15 mm2 15 mm2 x
Lsp Specimen gage length 50mm 50mm x
σy Yield strength - 800 MPa -
N Hardening exponent - 0.01 -
𝜀𝑓 Termination strain 3.5% 50% -
Figure 2.11 Specimen design used in the simulations of the Fast Jaws
The simulated stress strain-curves is shown in Figure 2.12A. The error in Figure 2.12B is the
difference between the calculated stress in the simulation and a perfect elastic response of the
specimen.
The conventional slack adapter with or without damping gave similar result for the simulated
stress-strain curve. The measured stresses were below the perfect linear response and an increased
damping gave a smaller error. The plot showed also that the error stabilised with increasing strain.
The linear elastic response of the specimen causes the force to increase constantly with an
increasing deformation. Due to the inertia of the load cell and upper grip the force measured with
the load cell constantly lacks after the actual load in the specimen, which causes the stabilised
error. The higher air damping caused a higher initial strain on the specimen before the specimen
was accelerated and the air damping thereby caused a lower stress “lack”.
For the fast Jaws system, the error varied in a different way. In this model, there was no lower
heavy grip to accelerate together with the specimen. The specimen was then loaded much quicker,
which caused the resonance wave in the stress measurement. In this case, the loading was too fast
and excited the resonance frequency of the simulated load cell/grip arrangement.
Figure 2.12A also showed that the beginning of the stress-strain curve is not linear. If the elastic
modulus was estimated with respect to ASTM 3039 [56], the recommended data in the strain range
0.1 – 0.3% would be obscured by the load cell resonance. The effect would be a too low estimated
elastic modulus. The error could be minimized by selecting a strain range, which was less affected.
However, that would require the materials to behave perfect linear elastic, before a fair comparison
could be made to quasi-static test, with the elastic modulus estimated in the recommended strain
range.
hspecimen
wspecimen/
wtab
tab specimen
Tabs Slack end with absolute stiffness sl
S
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A) Stress strain curves B) absolute error compared to perfect linear elastic
relationship
Figure 2.12 Simulation of stress strain curves for piston rod velocity of 20 m/s
Care should be taken when comparing elastic modulus obtained at high impact speed and obtained
at quasi-static conditions and these findings suggest that this kind of testing is unsuitable for
estimation of the elastic modulus.
Figure 2.13A shows the strain-rate vs strain plots for the simulations in Figure 2.12. For the
conventional slack adapter, the strain rate was rising through the entire deformation. This
happened since the lower grip never attained a constant velocity before the specimen reached its
fracture strain. This damping was enhanced by the rubber and air damping, which amplified the
problem due their lower accelerations of the lower grip.
A) Strain rate B) Force in the slack adapter. For the Fast Jaw
system the force was calculated in the long slender
part of the specimen.
Figure 2.13 Strain rate and force in the system.
The observation of non-constant strain rate was reported in literature for the conventional slack
0 1 2 3 0
200
400
600
800
1000
1200
Strain (%)
Spec
imen
str
ess
(MP
a)
Conventional slack adapter -With rubber damping -with rubber and air damping Fast Jaw system
0 0.5 1 1.5 2 2.5 3 -80
-60
-40
-20
0
20
40
60
80
Strain (%)
Ab
solu
te e
rro
r (M
Pa)
Conventional slack adapter -With rubber damping -with rubber and air damping Fast Jaw system
0 0.5 1 1.5 2 2.5 3 0
100
200
300
400
500
600
Strain (%)
Spec
imen
str
ain
rat
e (/
s)
Conventional slack adapter -With rubber damping -with rubber and air damping Fast Jaw system
0 0.5 1 1.5 2 2.5 3 0
100
200
300
400
500
600
Strain (%)
Forc
e (K
N)
Conventional slack adapter -With rubber damping -with rubber and air damping Fast Jaw system
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adapter, where a thicker rubber layer was suggested [57]. However, this modelling suggested that
a thicker rubber layer would damp the maximum strain rate further without making it constant. For
the Fast Jaws the strain rate increased faster due to the lack of inertial damping, however the strain
rate was not steady, but became a function of the oscillations introduced in the entire system.
Figure 2.13B shows the forces in the slack adapter vs. strain. For the conventional slack adapter
the maximum force were approximately the same, but the time the force peaked were different. In
the case of the combined rubber and air damping the air pressure had already compressed the
rubber so the final impact became harder than the case with the rubber alone. If the slack adapter is
too weak to withstand these forces, it would fail before any substantial forces are transferred to the
specimen. For the Fast Jaw system, the force in the slack length of the specimen is low, due to low
masses in the system.
A bouncing effect in the specimen displacement rate due to the non-rigid connection between the
specimen and the slack adapter has been reported in litterature [12, 13]. This bouncing effect was
caught by the model and is shown in Figure 2.14. The simulations revealed that the period of the
bouncing was long enough, so the specimens failed within the first bounce. In this case, the
specimen displacement rate becomes a function of the impact mechanism in the slack adapter and
not only a function of the piston rod velocity.
A) Bouncing effect for high and low grip mass. A
rubber layer is included as damping element for both
the high and low mass simulation.
B) Effect of additional air damping on the bouncing
effect
Figure 2.14 The bouncing effect because of no rigid connection between the slack adapter and the specimen.
In figure B additional damping was added by inclusion of air in the slack adapter.
2.4.1 Parameteric study
The model of the conventional slack adapter was used for a parameter study to examine the effect
of the load train parameters on the achieved strain rate in the specimen. The parametric study was
carried out by varying the different parameters of the model individually and calculating the
resulting average strain rate and the standard deviation in the interval [0.5𝜀𝑓: 𝜀𝑓] where 𝜀𝑓 were the
termination strain of the simulation. The parameters were varied according to
𝒙 = 𝑃𝑎𝑟 ∗ [0.1, 0.5, 0.75, 1, 1.5, 2, 3, 5] (2.17)
x was the vector of values used in the simulations; “Par” was the nominal value of the parameter
and the values in the square bracket were the multiplication factors to the nominal value. The
parameters, are given in Table 2.4 along with their nominal values.
0 10 20 30 40-200
-100
0
100
200
300
400
500
600Effect of lower grip mass
Time (ms)
Dis
pla
cem
ent
(mm
)
High mass - grip position
High mass - pistonrod position
Low mass - grip position
Low mass - pistonrod position
0 10 20 30 40-100
0
100
200
300
400
500
600Effect of lower grip mass
Time (ms)
Dis
pla
cem
ent
(mm
)
Grip position
Grip position with damping
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Table 2.4 Parameter description and their nominal values.
Parameter Description Nominal value
Lr Rubber thickness 3 mm
C1 Rubber shear stiffness 25MPa
Ls Slack adapter length 190 mm
As Slack adapter cross sectional area 452 mm2
Es Slack adapter elastic modulus 200 GPa
M1 Load cell grip mass 1.83 kg
M2 Moving grip mass 2.4 kg
M3 Piston rod mass 33 kg
Ag Cross sectional area, where air escapes in the slack adapter
5 mm2
The impact velocity was set to 2 and 20 m/s and the system pressure was set to 270 bars. The final
velocity at 270 bar pressure drop would then be 2 or 20m/s respectively. Both a linear elastic and
elasto plastic specimen were used for the simulations, and the parameters of the specimens are
given in Table 2.3. An elasto plastic specimen was included to compare how the varied parameters
may influence the specimen deformation differently dependent on the material response.
Figure 2.15 shows the normalised strain rate for the conventional slack adapter impacted at 20 m/s.
The x axis is the normalised factor of the parameter in question, e.g. a factor of two for the loadcell
grip mass M1 correspond to 2*M1.The y axis is the normalised strain rate. All strain rates were
normalized with the average strain rate obtained with nominal parameters. The three factors with
the largest influence on the average strain rate were the specimen length, the slack adapter stiffness
and the slack adapter cross sectional area. To maximize the average strain rate the absolute
stiffness of the slack adapter should be increased. Further the lower grip mass should also be
minimised.
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Figure 2.15 Parameter study of classical slack adapter setup. The largest effect on the strain
rates are found from the specimen length and cross sectional area of the slack adapter.
Impact velocity 20 m/s.
In Figure 2.16 the normalised strain rate is shown for an impact velocity of 2 m/s. The lower
impact energy changed the important parameters for the achieved strain rate. The strain rate
primarily depended on the specimen parameters due to the lower acceleration forces associated
with the lower velocity. The lower grip mass had a decreasing effect on the average strain rate
with decreasing weight, while for the 20m/s the lower grip mass had increasing effect for a
decreasing weight.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Normalised factor
No
rmal
ised
str
ain
rate
Air gap opening
Rubber thickness
Rubber shear stiffness
Slackadapter length
Slackadapter crosssectional area
Slackadapter stiffness
Specimen length
Specimen crosssectional area
Specimen stiffness
Upper grip mass
Lower grip mass
Impact mass
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Figure 2.16 Parameter study of classical slack adapter setup. Impact velocity 2 m/s.
For comparison an elasto plastic specimen were also simulated and the result is shown in Figure
2.17. Here the gage length were the dominant factor for the resulting strain rate. The impact mass
were seen to decrease the strain rate at higher masses, which is counter intuitive, but the reason is
because the acceleration length was kept fixed so the heavy mass could not reach full velocity
before impact. This was seen in the parameters studies for the linear specimens as well. If the
acceleration length is adjusted so the heavier impact mass can reach full velocity before impact the
influence disappears.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Normalised factor
No
rmal
ised
str
ain
rate
Air gap opening
Rubber thickness
Rubber shear stiffness
Slackadapter length
Slackadapter crosssectional area
Slackadapter stiffness
Specimen length
Specimen crosssectional area
Specimen stiffness
Upper grip mass
Lower grip mass
Impact mass
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Figure 2.17Parameter study for elastic plastic specimen. Impact velocity 20m/s
2.5 Summary
A model was presented to simulate the load train of a high speed servo hydraulic test machine. The
influence of the load train on the test results was examined and a series of conclusions could be
drawn from the modelling:
A linear elastic specimen cannot be deformed with a constant velocity in the high-speed
servo hydraulic test machine due to inertial effects.
There is no differential pressure to apply a force to a specimen when the piston rod
moves at constant velocity.
The deformation velocity transferred to the specimen was controlled by the mass of the
moving grip. In the case of the conventional slack adapter, which had a high moving
grip mass the velocity was constantly changing with the acceleration and deceleration of
the moving grip. For a Fast Jaw setup there was no lower moving grip, however the
transferred velocity was controlled by the elasticity of the slack length of the specimen
and in this case the velocity will neither be constant for linear elastic specimens.
For the conventional slack adapter the specimen failed within the first bounce of the
bouncing effect. For metal testing bouncing could lead to a load/ unload condition.
The rigid connection created with the Fast Jaw system, combined with the fast gripping
mechanism, lead to oscillations in the load cell before fracture of the linear elastic
specimen.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Normalised factor
No
rmal
ised
str
ain
rate
Air gap opening
Rubber thickness
Rubber shear stiffness
Slackadapter length
Slackadapter crosssectional area
Slackadapter stiffness
Specimen length
Specimen crosssectional area
Specimen stiffness
Upper grip mass
Lower grip mass
Impact mass
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Trapped air in the slack adapter caused high loading on the specimen before it was
accelerated. This could cause weak specimens to fracture before a high strain rate was
reached.
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3 Design and construction of a High- Speed Servo-Hydraulic Test Machine
The design and construction of a high-speed servo-hydraulic test machine (HS machine) is
described in this chapter. Chapter 2 describes the concept and working method of the HS machine
and this chapter describes the construction of a high speed servo hydraulic testmachin test
machine. Guidelines for the design process are developed, including the design of test specimens.
The machine was built with a limited budget so where possible existing equipment was rebuild
and reused for this purpose.
3.1 Design
The HS machine was divided into five design elements as shown in Figure 3.1, and the design
elements are described in the subsequent sections. Figure 3.1 also shows the conceptual setup of a
HS machine, where the main difference from a standard servo hydraulic test machine is the
introduction of the “lost motion unit”. The other main difference, not shown in the figure, was the
use of open loop control with no control of the piston rod position.
Figure 3.1 Design modules for design and construction of the
HS machine. Illustration from [1].
The performance requirements for the machine were set in Table 3.1.
achine frame
ylinder
anifold
alves
Accumulator system
Firing control
Open loop
losed loop
elocity control
Safety control
oad
Strain
Displacement
Images
Analog data -
collection
Triggering
Synchronisation
oad cell
Grips
ost motion unit
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Table 3.1 Performance requirements
Parameter Value
Static force >100 kN
Maximum velocity >20m/s
Acceleration length 0.2m
Working length 0.1m
Function Doubled ended piston rod. Must handle Tension and compression
Payload 20kg
The static force is the force delivered at maximum system pressure. The maximum velocity is the
velocity the piston rod achieves within the given acceleration length. The acceleration length was
set to 0.2m to maintain a stiff load train, but also to account for the weight penalty on the piston
rod from a long stroke. The working length is the remaining stroke when the piston rod has
reached constant velocity.
A payload of 20kg was added to the design weight to be accelerated to be sure the machine would
be able to accelerate the added grip system. Further, the machine should handle both tension and
compression loadings.
3.2 Frame and load unit
The hydraulic load unit consisted of the following parts
Cylinder
Manifold
Hydraulic valves
Accumulator system
The cylinder and manifold were designed in cooperation with the Swedish companies Industri
hydraulik SE (Manifold) and Argos SE (Cylinder).
Standard servo hydraulic test machines operate with a system pressure of 210 bars. If the system
pressure is increased to, for example, 280 bars then moving parts could be smaller while
maintaining the output force. Based on the availability of valves the system pressure was set to
maximum 280 bars with a minimum working pressure of 250 bars.
The initial cylinder design from Argos indicated a piston rod mass of about 35 kg. With a payload
of 20kg, the total accelerated mass became 55kg. As described in chapter 2 the pressure decreases
with increased flow through the valve and the acceleration force decreases. ombining Newton’s
second law and equation (2.16), yield the following relationship between the piston rod velocity �̇�,
and acceleration �̈�
�̈�𝑖 =𝐴𝑝
𝑀
𝑃𝑁
𝑉𝑁2 (𝑉𝑁
2 − �̇�𝑖2) (3.1)
Equation (3.1) was solved for 𝑥, �̇� and �̈� with at ab’s Ode solver. Vn was defined as the
velocity where all the pressure Pn in the valve was lost and Ap was defined as the pressurised area.
For this design Ap was fixed to 4778mm2. Figure 3.2 shows piston velocity vs. piston displacement
for this configuration for different velocities Vn. A configuration with Vn slightly above 20m/s
would reach close to a constant velocity of 20 m/s at 𝑥 = 0.2𝑚.
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Figure 3.2 Velocity profile of the piston rod as function
of the displacement. M = 55kg. Ap= 4778mm2. Pn =
280 bar.
A piston rod velocity of 20m/s corresponds to a flow rate of 5734 l/min for Ap = 4778mm2, and the
minimum flow requirement for the valve selection was set to 5800 l/min.
The total oil flow during the valve opening must be less than the total cylinder volume to finish the
piston rod acceleration before the piston rod reaches the end cushion. A linear relationship was
assumed between the opening of the valve and the flow, such that a 50% valve opening gave 50%
of the maximum flow. Assuming incompressibility of the oil, the piston rod velocity was
calculated from the pressurised area Ap and the flow rate Q as
�̇� =𝑄
𝐴𝑝
(3.2)
The acceleration length of the piston rod was set to xacc and the oil flow acceleration was assumed
constant. The maximum opening time was then calculated from
𝑡𝑜𝑝𝑒𝑛 =2𝐴𝑝𝑥𝑎𝑐𝑐
𝑄𝑚𝑎𝑥
(3.3)
Qmax is the maximum flow rate resembling the flow rate at constant velocity of the piston rod and
full pressure loss. Figure 3.3 shows relationship between valve maximum flow rate, the
acceleration length, the piston rod velocity, and the maximum valve opening time. With a flow
rate of 5800 l/min, the maximum opening time was found to be 21ms for an acceleration length
xacc of 0.2m.
No commercial valve rated for a maximum flow of 5800 l/min or higher is available, so two Moog
D665 valves was chosen. The Moog D665 valve is rated at 1500 l/min for a pressure drop of 10
bars, and with a maximum flow rate of 3600 l/min at 58 bar pressure drop. The response time to
100% opening is 12ms with a stub shaft spool configuration.
The Moog D665 valve can be used for open loop control as well as closed loop control. However,
the high flow rate make the D665 unsuitable for close loop control of the piston rod. Instead, a
MTS 256.09 servo valve was selected to drive the machine in closed loop control for positioning
the piston rod before testing. Ap = 4778mm2 corresponds to a force of 121kN at 250 bar and
135kN at 280 bar.
0 0.05 0.1 0.15 0.20
5
10
15
20
25
Piston rod displacement (m)
Pis
ton
ro
d v
elo
city
(m/s
)
Vn = 10m/s
Vn = 15m/s
Vn = 20m/s
Vn = 30m/s
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Figure 3.3 Valve selection chart – xacc is the acceleration length to full
flow rate. Ap = 4778mm2
The cylinder was designed for a dynamic stroke of 330mm, which covered both the required
acceleration length of 0.2m and the working length of 0.10 m. The full stroke was 608mm which
includes end cushions and clearance stroke for the inlet ports.
The accumulator system was designed to deliver an oil volume of 𝑉𝑅 = 𝐴𝑝 ∙ 330𝑚𝑚 = 1.6𝑙 from
a maximum pressure of P2 = 280 bars to a minimum pressure of P1 = 250 bars and at flow rates up
to 7200 l/min. The 7200 l/min was chosen to accommodate the full potential of the valves. With an
assumption of adiabatic expansion of the gas the required nominal accumulator volume V0 is given
by [58]
𝑉0 =
𝑉𝑅
(𝑃0
𝑃1)
1𝛾
+ (𝑃0
𝑃2)
1𝛾
𝐶𝑎 (3.4)
Ca is a correction constant with respect to the adiabatic process, γ . is the adiabatic constant of
the gas in accumulator and P0 is the gas pressure in the accumulator before oil pressure was
applied. With Ca = 1.4 and P0 = 180 bars, V0 is 36.4l. From these requirements, the Hydac SB330
H10 accumulator was selected. This accumulator has a nominal volume V0 of 10l and a rated flow
rate of 30l/s. Four accumulators were used on the pressure side in parallel connection to
accommodate the full flow of 4 ∙ 120l/s = 7200l/min and to obtain a total nominal volume of
40l. Additional four accumulators were placed at the return side of the cylinder to absorb the oil
and suppress pressure spikes in the system. The gas pre-pressure P0 was set to two bars. The
cylinder and manifold were designed with a port size of Ø80mm in areas where the full flow of
7200l/min would occur. Ø80mm equals an area of 5026mm2, so the port diameter was designed
larger than the pressurised area Ap = 4778mm2 in the cylinder.
The cylinder was designed with end cushions to smoothly stop the piston rod at the ends. Due to
the high piston rod velocities and the open loop control, it was impossible to stop the piston rod by
closing the valves, so the end cushions were designed to absorb the full kinetic energy of the
piston rod. The end cushions consisted of a chamber fitted with relief valves, where the piston rod
moved into at the end of the stroke. Inside the end cushions champers, the piston rod has to
compress the trapped oil out through the relief valves. Argos designed the end cushions to a
104MPa working pressure and an maximum pressure of 125MPa. With a pressure of 104MPa the
maximum dynamic force was calculated to
3000 4000 5000 6000 7000 8000 9000 100000
5
10
15
20
25
30
35
40
Max valve Flowrate Qmax (L/min)
Max v
alv
e o
penin
g t
ime (
ms)
3000 4000 5000 6000 7000 8000 9000 1000010
15
20
25
30
35
40
Pis
tonro
d V
elo
city (
m/s
)
Max valve opening time
Pistonrod velocity
xacc
: 0.1
xacc
: 0.2
xacc
: 0.3
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𝐹𝑑 = 104𝑀𝑃𝑎 ∗ 4778𝑚𝑚2 = 496𝑘𝑁 (3.5)
This load distributes though the machine and into the support and fundament of the machine, and
the dimensioning dynamic force was set to 500kN.
3.2.1 Machine Frame and support frame
A frame from an older MTS servo-hydraulic test machine was available to accommodate the
custom designed manifold and cylinder. Figure 3.4A shows a sketch of the frame and Figure 3.4B
shows the frame with the original cylinder removed. The frame was modified to support the new
manifold and cylinder, such that the impact forces are transferred from the cylinder to the frame
and the lab floor. The machine had a total height of 4.7 meters and had to be lowered into the lab
floor to obtain a feasible working height.
A) Drawing from the original documentation B) Frame with the original
cylinder removed.
Figure 3.4 Frame available to build a high speed servo hydraulic test machine
The entire machine frame and the cylinder and manifold were drawn as a 3D model, which was
used to design the modifications of the machine frame. Figure 3.5A shows the 3D model. A
support frame was constructed to create a platform for the test machine and to direct the impact
forces to the strongest part of the lab floor. Figure 3.5B shows the frame in the 3D model together
with a securing cage around the machine.
The weight of the designed manifold and cylinder was about 2.5 tons and the weight of the
machine frame was 2 tons. The maximum load on the support frame is 500kN + 25kN + 20kN =
545kN. Due to uncertainties in the actual dynamic pressure the design load was set to 650kN and
the safety factors in the steel construction standard DS412 [59] was used for dimensioning the
support frame.
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A) 3D model of the manifold and cylinder
installed in the frame.
B) 3D model of the full test machine
including support frame to fit the machine
into the lab floor.
Figure 3.5 3D model of the test machine
Figure 3.6A shows the installation of the complete manifold and cylinder into the machine frame.
Figure 3.6B shows the installation of the support frame into the lab floor and Figure 3.6C shows
the finished installation of the machine with control electronics next to the machine as well as a
security cage around the machine.
A) Installation in of the
actuator in the test machine
frame
B)Installation of the support frame in
the lab floor
C) The machine in place with
flooring and security cage and
the control electronics installed
at the left side.
Figure 3.6 Installation of the test machine
3.3 Control and DAQ system
The test machine was designed with both a closed loop and open loop control systems. The closed
loop control system was used for positioning the piston rod prior to firing, and was created with a
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MTS TestStar II Digital Controller. The closed loop controller provided also emergency stops and
safety chain features around the machine. The open loop control system was primarily used for
firing the Moog D665 valve for velocities above 1m/s, and to handle a number of other tasks, such
as data acquisition. The open loop control system performed the following task simultaneously
during a test:
Control of the Moog D665 valves with an analogue signal between -10 to 10 volt for
firing the machine.
Synchronised triggering of analogue data acquisition equipment (DAQ), and other
external equipment such as high speed cameras.
Keep track of the safety chain status from the TestStar II controller.
Set the closed loop system in the TestStar II controller on idle, such that the closed loop
control did not interfere with operation of the main valves.
A program was written in LabView to handle these tasks, and the system simultaneously handled
analogue in- and output, and digital in- and outputs. The hardware part consisted of two National
Instrument 6115 multifunctional PCI 12bit DAQ boards with an analogue sampling frequency of
maximum 10MHz synchronized on eight channels. The boards were installed directly to the PCI
bus in a high performance desktop computer. The DAQ board was programmed to work in two
modes: a slow sampling mode with streaming to the PC memory for showing current data during
setup of the machine, and a transient mode where the DAQ board’s internal memory was set up as
a ring buffer, so data was only streamed to the board memory during a test. The latter mode
prevented the PCI bus to be flooded with data at maximum sampling frequency. The board
memory could hold about 4.3M samples per channel and the memory control was configured to
allow an arbitrary distribution of the memory around the trigger. The data before the trigger was
pre-trigger data and the data recorded after the trigger was post-trigger data. When the board
memory was full after triggering, the data was transferred to the PC memory. Table 3.2 provides
the simplified firing sequence of the system.
Table 3.2 Simplified firing sequence. When the user asked to fire, step 5 allowed the used to abort the
operation.
Step Description
1 Prepare TestStar II controller for idle mode.
2 Prepare analogue output sequence based on the selected velocity and load into output buffer of the board.
3 Put analogue output into a firing position and await firing trigger.
4 Stop the slow sampling mode of the DAQ board, reconfigure to transient sampling, and begin ring buffer
streaming.
5 Ask user to fire or abort (if the safety chain of the TestStar II controller is still unbroken!)
6 If firing is accepted, put TestStar II on idle.
7 Send trigger to the analogue out, to open the valves and move the piston rod.
8 Wait until analogue sampling has finished and transfer recorded data to the PC memory for saving and post processing of the data. Put TestStar II to active mode.
9 Set DAQ board back to slow sampling mode.
The complete firing sequence included several controls signals sent back and forth to Moog valves
and the TestStar II controller to ensure all steps were completed correctly.
A test at a target velocity of 20 m/s would complete in less than 30ms and would leave no
possibility for human interaction for triggering of the DAQ board. Instead, the trigger system was
designed to be fully automated and to work with external equipment as well. The trigger to fire the
machine was activated via a software button such that only the operator could fire the machine.
The DAQ equipment and external equipment could be connected to this trigger, or they could
work with their own triggers as for example the build in level trigger circuit on the NI 6115 board.
A level trigger, triggers when its analogue channels reaches a pre-set level. The external
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equipment, as for example a high-speed camera was connected to these triggers and synchronise
with the DAQ board. The concept of variable trigger setting and possible trigger dependency is
visualised in Figure 3.7.
Figure 3.7 Trigger setup
This arrangement provides a flexible setup where data acquisition could be triggered in any
possible way and allows for time synchronisation of external equipment.
Before the machine could be fired, the correct voltage signal was generated for the valves. The
signal generation was based on a calibration of the machine. The calibration was performed by
opening the valves with a given constant voltage signal, and measuring the resulting steady
velocity. Figure 3.8A shows the calibration curve with velocity as function of signal voltage.
Figure 3.8B shows a sketch of the actual signal sent to the valve.
A) Velocity calibration curve B) Sketch of valve drive signal. Signal was ramped to
minimize oscillations in the system.
Figure 3.8 Velocity calibration and used valve signal
The LabView program had three user interfaces. The main user interface allowed the user to select
velocity and setup, and to see the status of the DAQ channels and the status of the prerequisites to
fire the machine. The window also contained a graph to visualise the acquired analogue data after
a test. The DAQ interface allowed the user to setup the analogue channels and to set the ring
buffer setting and the trigger routes. The saving window allowed the user to select which data to
save.
A Photron APX-RS camera was available for use with the machine and the cameras control was
integrated into the program for automatic synchronisation of images and analogue data.
evel trigger
Trigger out
Trigger out
ternal trigger
ternal trigger
Software button
0 2 4 6 8 100
5
10
15
20
25
30
Control Voltage (V)
Fin
al c
on
stan
t ve
loci
ty (m
/s)
t
volt
tend
selected
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3.4 Control electronics
All the electronics of the machine was assembled in control tower. A custom connector box was
designed and built, which connected the NI 6115 board with the MTS controller, the valves, and
the sensor amplifiers. The connector box was also equipped with BNC connections for distributing
the trigger signals and sensor signals, and further it did also provide a power supply for the Moog
valves. The installation of the control tower required custom cabling of all the components. Figure
3.9 shows the control tower, which also includes a PLC control for the pump. All PLC controls
were provided by a technician.
Figure 3.9 Control electronics
3.5 Sensors
The machine was equipped with a Sentech HydraStar displacement transducer (Linear Variable
Displacement Transducer, LVDT) mounted in the piston rod for monitoring of the piston rod
position. This LVDT was chosen due to its specialised design for high-speed motions.
The machine was also equipped with a Kistler 9071 piezo electric load cell. Piezo electric load
cells are designed to measure compressive forces, but here the load cell was implemented into the
upper grip structure such that tensile forces could be measured [60]. This was done by preloading
the load cell and then let tensile forces reduce the compressive. The amount of preload determines
the load capacity of the setup. The preload was limited by the absolute compressive load capacity
of the load cell and for this setup, the preload was limited to 150kN by the grip design. The
implementation of the load cell in the upper grip system is described in section 3.7.
The machine was equipped with a three-channel FYLDE HT379TA broadband strain gage
amplifier. Broadband amplifiers are normally rated with a cut-off frequency and their slew rate
referred to input (RTI) or referred to output (RTO). The slew rate is the maximum rate of change
the amplifier can sustain, normally given in the units V/µs. For example when the slew rate is 8
V/µs RTO it means the amplifier can maximum change its output with 8 V/µs. The FYLDE HT
379TA had a slew rate of 8V/ µs RTO and the selection guide in Appendix E reveals that a slew
rate of 1V/ µs or more is admissible for the high-speed servo hydraulic test machine.
3.6 Specimen design
The test specimen is an integral part of the load train and the design of the specimen influences the
load train design and vice versa. In this case, the specimen design was considered first, but the
oadcell amplifier
Strain gage amplifier
P control for the pump
onnection bo and
power supply to the
valves
ontrol pod for TS
closed loop controller
P for TS controller
P for DAQ equipment
and valve control
TS controller ots of wires
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limitations in the grip design were kept in mind. Only tensile testing was considered in this work.
Dogbone specimens are normally used for tensile testing of metallic materials, where the narrow
part promotes the failure position. This design is not suitable for FRP materials, as the fibres area
required to be cut in the loading direction. Instead, straight sided specimens are used [2], which
insures all fibres in the specimen are loaded. However, the straight-sided design complicates the
promotion of failure in the gage length and it becomes important to ensure a smooth load transfer
between the grip and specimen to avoid premature failure within the grips. Specimens can be made
with bonded tabs, loose tabs as emery cloth, or no tabs. Bonded tabs are strongly recommended for
highly unidirectional specimens [56]. If segregated wedges are used in wedges grips, the tabs also
prevents fibre damage, when the grip tightens around the specimen and ‘bite’ into specimen. With
bonded tabs the load transfer happens through the bond line, and the premature failure could then
happen in the tabs (sliding), the bond line (failure) or in the substrate for the bonding.
Straight- sided coupons with bonded tabs were used in this work. The design of the bond line was
done in accordance with ASTM 3039 [56] by designing the Ltab according to
2𝐺𝑠𝑢𝑊𝑡𝑎𝑏𝐿𝑡𝑎𝑏 > 𝜎𝑡𝑢ℎ𝑠𝑝𝑒𝑐𝑖𝑚𝑒𝑛𝑊𝑠𝑝𝑒𝑐𝑖𝑚𝑒𝑛 ⟺
2𝐺𝑠𝑢𝐿𝑡𝑎𝑏 > 𝜎𝑡𝑢ℎ𝑠𝑝𝑒𝑐𝑖𝑚𝑒𝑛 ⟺
𝐿𝑡𝑎𝑏 >ℎ𝑠𝑝𝑒𝑐𝑖𝑚𝑒𝑛
2
𝜎𝑡𝑢
𝐺𝑠𝑢
(3.6)
𝜎𝑡𝑢 is the ultimate tensile strength of the specimen and 𝐺𝑠𝑢 is the ultimate shear strength of the
glue. Equation (3.6) shows the tab length solely depended on the specimen failure strength, the
bond line shear strength, and the specimen thickness. The specimen should be thin with long tabs
and a strong glue to promote failure in the gage section of the specimen. Figure 3.10A shows and
overview of the tab setup for a specimen with tabs on both sides, and Figure 3.10B shows equation
(3.6) plotted for different values of the parameters.
A) Exploded view of tabs around one of the specimen.
B) Maximum thickness of specimen as function of the
tab length and relative strength between specimen
and glue.
Figure 3.10 Design of specimen and tabs
A substantial amount of trial and error was required in this work to establish a working specimen
design. The trial and error were related to uncertainties in the specimen tensile strength and the
bond-line shear strength. The tensile strength increased in many cases with strain rates, and tab
designs which worked at quasi-static strain rates, did not work at high strain rate. The bond
strength was also indicated to change with strain rate. The solution was to decrease the design
bond strength with 50% and design the specimen accordingly.
For quasi axial specimen it was found that the outer layer, which was glued to the tab had to be be
tab
Wtab
Wspecimen
hspecimen Tab
Glue area
0 50 100 1500
1
2
3
4
5
6
7
8
9
Relative strength<tuGsu
Max
imu
m s
pec
imen
th
ickn
ess
(mm
)
50 mm tab length
60 mm tab length
75 mm tab length
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the strongest layer with respect to the loading direction. Off axis, layers were found to fracture in
tab area, if they were placed in contact with the tab. An example is given in Figure 3.11A where a
90° and 45° plies were visible on the tabs, which was ripped off during the test. Figure 3.11B
shows an example with a UD 0° Eglass/LPET specimen tested at 20m/s where the bond lines
failed. The glue was still attached to the aluminium tabs, so failure happened in the glue/specimen
interface.
A) Example of a tab failure for a multidirectional
specimen with an angled outer layer
B) Example of bond line failure for a UD 0°
specimen
Figure 3.11 Examples of tab failures
3.6.1 Gage length
As discussed in chapter 2 the gage length has high influence on the achievable strain rate for a
straight-sided test specimen. ASTM 3039 suggests a gage length of 138mm for a 0° layup.
However long gage length requires high impact velocities and it was preferable/required to reduce
the gage length if a certain strain-rate regime had to be accessed. The drawback of the smaller
gage lengths is a higher vulnerability to misalignment and a less representative nature of the bulk
material.
3.7 Load train
The load train includes the load cell, the grips to fixate the specimen and the slack adapter. The
conventional slack adapter design from chapter 2 was selected. Standard grips, both hydraulic and
mechanical, were found to be too heavy to
be used in as their weight easily exceed 15-20kg for 100kN design. Instead, a set of highly
optimized and customized mechanical wedge grips were designed. High-strength tough aluminium
was used for the housing of the grips, and the wedges were made from hardened steel. The
maximum specimen length inside the grip was set to 60mm as a compromise between weight and
maximum tab length on the specimens. The grip consisted of two parts: a static, and a movable.
The movable part was designed with a wedge shaped cavity to house the wedges. The two parts
was fastened to each other with bolts. When the bolts were tighted manually, the wedge was
forced together and the specimen was clamped. The maximum specimen thickness was 9.7mm
including tabs. Figure 3.12A shows the upper grip mounted in the test machine and Figure 3.12B
shows a static finite element simulation of the grip with the specimen pulled at 150kN. The used
aluminium had a yield strength of 620MPa.
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A) Upper grip assembly mounted on the machine B) Von Mises stress in a static Finite element
simulation of the upper grip assembly with 150kN of
load applied to the specimen. Specimen had full
contact to the wedges.
Figure 3.12 The upper grip system and the finite element simulation of the system
The upper grip assembly also accommodated the piezo electric load cell, where the load cell was
mounted between the upper grip and the crosshead of the machine frame. The crosshead had a
through hole in the centre and a tie rod was mounted through the hole to pull the upper grip up
against the crosshead as shown in Figure 3.13. The tie rod was then used to preload the load cell.
Calibration of the load cell was carried out as the tie rod shunted the load cell.
Figure 3.13 Upper grip assembly and the mounting to the crosshead
The lower grip was designed similar to the upper grip assembly and made use of the same
components. The weight of a full grip assembly was about 1.5kg. The lower grip was connected to
the piston rod with the slack adapter. The slack adapter was designed as a long slender bar, which
was screwed up into the lower grip. The piston rod of the cylinder was made hollow to save
weight and the space was used to install the slack adapter down into the piston rod. The plug was
designed with a hole where the slack adapter could pass through.
Figure 3.14 shows the parts used for the load train including wedges and equipment for aligning
the load train. Attachment equipment for the reference load cell for calibration is also shown. The
slack adapter assembly is shown in the left with the lower grip attached.
The wedges were made from hardened steel to accommodate hard materials to be gripped. The
surface was segregated to allow a mechanical interlock to be created to the tabs of the test
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specimens. Two different patterns were used, with the coarse one for soft tabs where deep
mechanical interlock was needed to create a strong connection.
A) Parts for the load train B) Wedges – with different patterns
Figure 3.14 Parts for the load train. The upper grip assembly is not shown.
3.8 Issue with grips
It was found that the highest forces in the load train occurred when the lower grip impacted the
piston rod upon deacceleration of the piston rod. Due to the requirement of a stiff slack adapter
there was no room for a large deacceleration length, and the impact was normally damped with a
rubber layer. Figure 3.15A shows a failed rubber damper, which caused the grip in Figure 3.15C to
fail under its own weigh during impact. The rubber damper was changed to a thick PUR damper,
which was custom casted from a Shore 72D polyurethane base. The lower grip was further
modified to remove stress concentration critical to the deacceleration loading. The modifications
solved the problem.
ower grip
Slack adapter
Slack adapter plug ,
to be mounted in
piston rod
Impact damper
Slack adapter retainer.
quipment for
calibration of
the loadcell
Alignment tool
for aligning the
load train
Torque wrench
for tightening the
grips
Wedges
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A) Failed impact damper. Lose
undamaged damper shown next
to the piston rod.
B) Updated impact damper
(Green) casted from PUR shore
72D.
C) Broken and updated grip
Figure 3.15 Impact damper for deaccleration of the lower grip.
3.9 Summary
A high-speed servo hydraulic test machine was designed, constructed and equipped with a full set
of sensors to monitor test specimens during testing. The work also included constructing the full
control system as well as a support frame to integrate the machine into the lab.
The work included design of a conventional slack adapter and custom grips and equipment to
mount the piezo electric load cell. The load cell and grips were designed to maximum tensile loads
of 100kN. The slack adapter system was aimed at testing fibre-reinforced composites with respect
to the choice of a conventional slack adapter and rough segregated wedges for gripping soft
materials. The velocity calibration in Figure 3.8A showed the machine was able to reach velocities
in excess of 28m/s.
Department of Mechanical Engineering
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4 Characterisation of fibre reinforced materials at medium strain rates
This chapter present an examination of the medium strain rate behaviour of three fibre-reinforced
polymers (FRP). An Eglass/Epoxy, an Eglass/Lpet and Carbon/PA6 FRP was tensile tested from 1
/s to 300/s with the high-speed servo-hydraulic test machine described in chapter 3. The findings
was used as validation cases for the model presented in chapter 2 and as input data for the RESIST
project mentioned in the preface.
4.1 Method
The three materials were tensile tested at three loading velocities 1, 7.5 and 20 m/s, in two loading
direction and with two layups. The two lay-up were a UD and a Quattro axial lay-up (named MD).
The two loading direction for the UD configuration were 0°, along the fibres, and 90°, transverse
to the fibres. The MD lay-up had no dominant fibre direction and the test direction was just named
“Tensile”. Table 4.1 details the test combinations matrix. Each test series is identified with a code
as “ J ” and covers a combination of velocity, loading direction, and lay-up. Five
specimens were tested for each test series and the test matrix Table 4.1 covers in total 125 tests.
Specimen size was given as total size and the gage length LS. The general specimen design is
shown in Figure 4.1 and the details of the specimen materials, fabrication methods and tab size is
given in Table 4.2. Figure 4.2 shows specimens before testing.
Table 4.1 Test matrix. IP = In plane.
Material No Description Deformation
rate (m/s)
Specimen
(mm)
Lay-up Loading
Results
Eglass/E
poxy
CRBJ20B
CRBJ20C
CRBJ20D
IP
Tension 0°
1
7.5
20
Straight
150x15x1
LS = 50
[0]4 Tension along
1 axis
E11t,
ν12t,
X11t
Eglass/Epoxy
CRBJ21B CRBJ21C
CRBJ21D
IP Tension 90°
1 7.5
20
Straight 150x25x4
LS = 50
[0]4 Tension along 2 axis
E22t
ν21t,
X22t
Eglass/E
poxy
CRBJ21E IP
Tension 90°
1
Straight
150x25x4 LS = 100
[0]4 Tension along
2 axis
E22t
ν21t,
X22t
Eglass/E
poxy
CRBJ10B
CRBJ10C
CRBJ10D
IP
Tension
1
7.5
20
Straight
150x25x4
LS = 50
[0/-45/90/45/-
45/90/45/0]3
Tension along
x- axis
Exxt
νxyt,
Xxxt
Eglass/Lpet
CRBB20B CRBB20C
CRBB20D
IP Tension 0°
1 7.5
20
Straight 150x15x1
LS = 50
[0]4 Tension along 1 axis
E11t,
ν12t,
X11t
Eglass/L
pet
CRBB21B
CRBB21C CRBB21D
IP
Tension 90°
1
7.5 20
Straight
150x25x4 LS = 50
[0]8 Tension along
2 axis
E22t
ν21t,
X22t
Eglass/L
pet
CRBB10B
CRBB10C
CRBB10D
IP
Tension
1
7.5
20
Straight
150x25x4
LS = 50
[0/45/90/-45]s Tension along
x- axis
Exxt
νxyt,
Xxxt
C/PA6 CRBF20A CRBF20B
CRBF20C
IP Tension 0°
1 7.5
20
Straight 150x15x1
LS = 50
Winding Tension along 1 axis
E11t,
ν12t,
X11t
C/PA6 CRBF21A IP 1 Straight Winding Tension along E22t
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CRBF21B
CRBF21C
Tension 90° 7.5
20
150x25x4
LS = 50
2 axis ν21t,
X22t
Figure 4.1 General specimen design
A) UD layup, 0° loading direction. B) UD layup, 90° loading
direction.
C) Quattro axial layup – Loading
direction: x.
Figure 4.2 Specimens before test for an impact velocity of 20 m/s
Table 4.2 Test coupons
and materials
Table header Eglass/Epoxy Eglass/Lpet Carbon/PA6
Matrix ARALDITE LY 564/ ARADUR 917
LPET PA6
Fibre SAERTEX S14EU990-
00940-T1300-499000
TufRov 1200 Tex 4588 Tenax 12k X011 HTS 40
Production method Vacuum infusion. Specimen
cut watercooled diamond saw
Commingled yarn -
Vacuum consolidation Specimen cut water cooled
diamond saw
Commingled yarn -
Winding. Specimen cut watercooled
diamond saw
UD 0°
Tabs (Ltab xWtab) 60x15mm [0/90]
Eglass/Epoxy
60x15mm [0/90]
Eglass/Epoxy
60x15mm [0/90]
Eglass/Epoxy
UD 90°
Tabs (Ltab xWtab) 40x25mm [0/90] Eglass/Epoxy
40x25mm [0/90] Eglass/Epoxy
40x25mm [0/90] Eglass/Epoxy
Quattro axial
Tabs (Ltab xWtab) 60x25mm [0/90]
Eglass/Epoxy
60x25mm [0/90]
Eglass/Epoxy
All specimens were equipped with tabs glued with Scotch DP460 Epoxy glue and all tabs were
made from standard Eglass/epoxy [0/90]x tab material. Tab length was maximized with respect to
the specimen description in chapter 2. Department of Wind Energy (DTU) tested the materials
under static condition and all result mentioned as “static” results in this chapter comes from
hspecimen
wspecimen/
wtab
tab
specimen
Tabs Tabs
S
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Toftegaard et al. [62, 63].
Load was measured with the Kistler 9071 Piezo electric load cell and the load was converted to
stress by dividing the load with the initial cross sectional area of the test specimen. A strain gage
was glued to the gage area with cyanoacrylate for measuring strain. Two test specimens in each
test series were equipped with a 0°/90° Ω strain gage with 6mm gage length (Vishay EA-06-
125VA-350) for measuring Poisson’s ratio. The remaining specimens were equipped with a
Vishay EA-06-250BF- Ω strain gage, also with 6mm gage length. In addition, a high-speed
video was captured of the test specimen with a single Photron APX-RS high-speed camera. All
test specimens were spray painted with a speckle pattern to employ the images for 2D Digital
Image Correlation (DIC) with the DIC code ARAMIS 2D. Prior to testing, the camera were
calibrated for scale, and lens distortions. The set-up details are given in Table 4.3 and examples of
speckle patterns are shown in Figure 4.3 Examples of speckle patterns. ARAMIS creates a
displacement field for the specimen, which is converted into a strain field. The strain field
consisted of a strain tensor estimated with a least square method in each facet from the eight
surrounding facets. The strain measure for the characterisation of the test coupon was taken as the
average strain of all calculated facets over the gage area.
Table 4.3 Digital image correlation setup
Technique used 2D Image correlation, Calibrated
Subset size 15 x 15px – 19 x 19px
Facet step 13 px
Field of view 144 x 640px - 96 x 256px
Frame rates 10000 – 70000 FPS
Shutter time 2-10µs
Camera PHOTRON APX-RS
Lens 50mm
Light Dedocool
Figure 4.3 Examples of speckle patterns.
4.2 Results
The measured stress-strain data were used to construct the stress-
strain curve. Further, the elastic modulus, the failure stress, the failure strain, and strain rate were
calculated as given in Table 4.4. For each parameter in each test series, the 95% confidence
interval was calculated assuming a normal distributing with n-1 degrees-of-freedom where n was
the number of test specimens in the test series. The 95% confidence interval is written in
parenthesis after the mean value.
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Table 4.4 Parameter estimation
Parameter Method Data source
Elastic moduli Least square regression in the 0.2% - 0.5% strain interval. The interval may
vary slightly dependent on data quality
Strain gage and DIC. Average strain calculated before
regression.
Failure stress Maximum load divided by cross
sectional area
Kistler load cell
Failure strain Strain at maximum load. If there is no strain data, strain data is
extrapolated.
Strain gage/DIC
Strain rate Calculated as the time derivate of strain
measurement. The reported strain rate is calculated as the average strain rate
over the last 50% of strain in the
specimen before failure.
Strain/DIC
Poisson’s ratio Least square regression over linear part of inplane strains – strain interval
maximized to minimize noise.
DIC – average x and y strain of the estimated strain tensors
/ Strain gage where available.
Figure 4.4 shows examples of specimens after test for all test velocities.
1m/s
7.5m/s
20m/s
A) UD layup.
Loading direction: 0°.
B) UD layup.
Loading direction: 90°.
C) MD layup.
Loading direction: x.
Figure 4.4 Specimens after test for an impact velocity of 20 m/s
There was sign of higher splitting of the specimen for the UD 0° specimen at higher velocities.
The 90° did not have any significant changes in its fracture pattern and the same was the case for
the MD specimens. Figure 4.5 gives examples of stress strain curves for UD 0° at all impact
velocities.
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A) Impact velocity: 1 m/s B) Impact velocity: 7.5 m/s C) Impact velocity: 20 m/s
Figure 4.5 Example of stress strain curves for Eglass/Epoxy 0° loading direction
All estimated parameters are given in Table 4.6 to Table 4.9. All results in the columns “static” are
taken from [62, 63].
Table 4.5 Elastic modulus (GPa)
Static 1m/s 7.5 m/s 20 m/s
Eglass/Epoxy 0° 45.3 (1.3) 42 (3.42) 42.5 (2.0) 41.2 (2.1)
Eglass/Epoxy 90° 12.7 (0.8) 13.5 (3.4) 13.6 (1.88) 12.2 (2.2)
Eglass/Epoxy MD 22.1 (0.6) 19.6 (0.9) 20.1 (1.9) 18.2 (2.8)
Eglass/Lpet 0° 39 (0.3) 36.2 (3.3) 37.3 (1.8) 34.3 (4.1)
Eglass/Lpet 90° 9.5 (0.2) 9.2 (2.4) 10.5 (2.4) 8.6 (2.3)
Eglass/Lpet MD 19.7 (0.3) 15.0 (1.1)
Carbon/PA6 0° 100 110 (10.8) 103 (7.3) 97.6 (7.7)
Carbon/PA6 90° 8.9 6.2 (0.4) 5.7 (2.0) 6.75
Table 4.6 Failure stress (MPa)
Static 1m/s 7.5 m/s 20 m/s
Eglass/Epoxy 0° 1041 (31) 1095 (42) 1274 (186) 1485 (84)
Eglass/Epoxy 90° 29 (3) 26.3 (11.3) 34.3 (2.01) 32 (5.1)
Eglass/Epoxy MD 387 (8) 481 (23) 554 (11) 616 (31)
Eglass/Lpet 0° 789 (21) 903 (84) 882 (69) 788 (82)
Eglass/Lpet 90° 33 (2) 27.1 (18.4) 33.3 (37.8) 30.4 (6.0)
Eglass/Lpet MD 337 (9) 315 (29)
Carbon/PA6 0° 931 1569 (107) 1869 (205) 1816 (184)
Carbon/PA6 90° 70 46.6 (12.3) 39.4 (19.4) 48.25
Table 4.7 Failure strain (%)
Static 1m/s 7.5 m/s 20 m/s
Eglass/Epoxy 0° 2.9 (0.8) 2.77 (0.38) 3.85 (0.16) 4.41 (0.25)
Eglass/Epoxy 90° 0.22 (0.03) 0.19 (0.09) 0.28 (0.05) 0.54 (0.07)
Eglass/Epoxy MD 2.7 (0.2) 3.56 (0.19) 3.91 (0.09) 4.15 (0.68)
Eglass/Lpet 0° 2.2 (0.1) 2.65 (0.15) 3.12 (0.69) 3.57 (0.87)
Eglass/Lpet 90° 0.39 80.03) 0.38 (0.14) 0.32(0.4) 0.28 (0.13)
Eglass/Lpet MD 2.65 (0.184) 3.58 (1.00)
Carbon/PA6 0° 1.0 1.31 (0.10) 1.59 (0.21) 1.87 (0.17)
Carbon/PA6 90° 0.85 0.63 (0.2) 0.38 (0.07) 0.63
0 1 2 30
200
400
600
800
1000
1200Stress Strain curves
Strain (%)
Stre
ss (M
Pa)
CRBJ20B-01
CRBJ20B-02
CRBJ20B-03
CRBJ20B-04
CRBJ20B-05
0 1 2 3 4 50
200
400
600
800
1000
1200
1400
1600Stress Strain curves
Strain (%)
Stre
ss (M
Pa)
CRBJ20C-01
CRBJ20C-02
CRBJ20C-03
CRBJ20C-04
CRBJ20C-05
0 1 2 3 4 50
200
400
600
800
1000
1200
1400
1600Stress Strain curves
Strain (%)
Stre
ss (M
Pa)
CRBJ20D-01
CRBJ20D-02
CRBJ20D-03
CRBJ20D-04
CRBJ20D-05
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Table 4.8 Poisson’s ratio (-)
Table header Static 1m/s 7.5 m/s 20 m/s
Eglass/Epoxy 0° 0.27 (0.013) 0.26 (0.05) 0.30 (0.04) 0.30 (0.01)
Eglass/Epoxy 90° 0.077 (0.008) 0.08 0.05 0.06
Eglass/Epoxy MD 0.301 (0.003) 0.27 (0.01) 0.27 (0.03) 0.31 (0.07)
Eglass/Lpet 0° 0.332 (0.013) 0.33 (0.04) 0.29 (0.02) 0.28 (0.07)
Eglass/Lpet 90° 0.083 (0.004)
Eglass/Lpet MD 0.336 (0.011) 0.30 (0.05)
Carbon/PA6 0° 0.32 0.36 (0.04) 0.36 (0.11) 0.31(0.08)
Carbon/PA6 90° 0.036 0.018 0.023
Table 4.9 Average strain rate (/s)
Table header Static 1m/s 7.5 m/s 20 m/s
Eglass/Epoxy 0° 2.6 (1) 78 (39) 298 (38)
Eglass/Epoxy 90° 3.6 (2.3) 3 (1.24) 4.8 (3.8)
Eglass/Epoxy MD 2.4 (1.1) 94 (13) 211 (75)
Eglass/Lpet 0° 2.8 (0.6) 121 (20) 258 (126)
Eglass/Lpet 90° 7.3 (1.4) 2.3 (1.4) 2 (1.8)
Eglass/Lpet MD 3.4 (0.7)
Carbon/PA6 0° 3 (0) 90 (20) 135 (24)
Carbon/PA6 90° 6 (4.9) 2.8 (2.4) 6.5
The data in Table 4.9 show that the strain rate for the 90° specimens was low even at high impact
velocities, since the specimens were broken before the slack adapter completely engaged the
specimen. The effect could be seen for the 0° and MD specimens where the strain – strain rate
curves in Figure 4.6 shows an initial ramp with low strain rates up to a strain level between one
and two percent, before the strain rate increased. The 90° fracture strains are below 1% so they
failed in this preload range. The strain rate curves also showed that it was not possible to achieve a
constant strain rate.
A) 7.5m/s Impact velocity B) 20 m/s Impact velocity
Figure 4.6 Example of strain - strain rate curves. The non-constant strain rate was found for all tests!
Figure 4.7 show the graphs of data in Table 4.6 to Table 4.8. For each test series, the average
parameter values were normalised with the mean value of the static test and plotted against the
0 0.5 1 1.5 2 2.5 3 3.50
50
100
150
200
Strain (%)
Stra
in r
ate
(/s)
CRBJ20C-01
CRBJ20C-02
CRBJ20C-03
CRBJ20C-04
CRBJ20C-05
0 1 2 3 40
50
100
150
200
250
300
350
400
Strain (%)
Stra
in r
ate
(/s)
CRBJ20D-01
CRBJ20D-02
CRBJ20D-03
CRBJ20D-04
CRBJ20D-05
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average strain rate for the test series.
Figure 4.7 Averaged normalised Failure stress, Failure strain, Elastic modulus and poisons ratio as function
of strain rate
10-4
10-3
10-2
10-1
100
101
102
103
0
0.5
1
1.5
2
2.5
Strain rate (/s)
>=>
static
10-4
10-3
10-2
10-1
100
101
102
103
0
0.5
1
1.5
2
2.5
Strain rate (/s)
E=E
static
10-4
10-3
10-2
10-1
100
101
102
103
0
0.5
1
1.5
2
2.5
Strain rate (/s)
" tu=" t
usta
tic
10-4
10-3
10-2
10-1
100
101
102
103
0
0.5
1
1.5
2
2.5
Strain rate (/s)
<tu=<
tusta
tic
Eglass/Epoxy - 0
Eglass/Epoxy - 90
Eglass/Epoxy - MD
Eglass/Lpet - 0
Eglass/Lpet - 90
Eglass/Lpet - MD
Carbon/PA6 - 0
Carbon/PA6 - 90
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Figure 4.6 present examples of strain - strain rate curves and Figure 4.8 present an example of the
curve from where poisons ratio is estimated.
Figure 4.8 Example of curve from which poisons
ratio is estimated as the slope of the curve. This curve
is for UD 0° Eglass/Epoxy at 20m/s.
4.3 Discussion
Figure 4.7 indicates that the failure stress and strain are more affected of strain rate than the elastic
parameters (the elastic modulus and Poisson’s ratio). Both the failure stress and strain increase
with strain rate while the elastic parameters are unaffected. Both the Eglass/Epoxy UD 0° and MD
laminates showed a positive trend with strain rate whereas the UD 0° Eglass/Lpet first increased
and then dropped at the highest strain rates. The 90° specimens could not be tested at higher strain
rate due to limitations in the load train. The Carbon/PA6 dropped in fracture stress and strain at
higher strain rates whereas the Eglass/Epoxy and Eglass/Lpet stayed at the same level. It should be
noted that the 0° Carbon/PA6 shows a high degree of rate dependency from static to 1/s. The static
measurements were done in a former project [64] while the new specimen was produced for this
project. After the test were carried out, it was noted that the specimens for static testing was
produced with a Tenax STS (Standard tensile strength) [65] carbon fibre while the new specimens
was manufactured with a Tenax HTS (High tensile strength) carbon fibre [64]. The HTS fibre has
a typical strength of 4300MPa while the STS fibre has a typical strength of 4000 MPa. The low
static value of around 1GPa in failure strength cannot be explained solely from the use of a
different fibre. The static measurement puts the Carbon/PA6 in level with the Eglass/Epoxy
material. The apparent high strain rate effect indicated here is doubtful because of the low static
measurement. Only small increase in fracture strength is reported in literature for carbon materials
[24, 26].
The elastic modulus was not significantly affected by strain rate, but the estimates at higher strain
rates had high scatter. The ASTM D3039 standard [56] prescribed a strain interval from 0.1 to
0.3% strain for estimation of the elastic modulus, but the first amount of strain could be disturbed
by resonance in the load cell as described in section 1.5. Further, the strain – strain rate plots
showed a low strain rate in this strain range. If the strain rate and elastic modulus were estimated
at higher strain rate, any nonlinearity in the material behaviour would affect the calculated elastic
modulus. For these reasons it is not possible to say if the elastic modulus is unaffected by the
strain rate. Poisson’s ratio was estimated in the same range as the elastic modulus and the same
arguments apply as for the elastic modulus. However, Figure 4.8 shows that Poisson’s ratio (slope
0 0.5 1 1.5 2 2.5 3 3.5-1
-0.8
-0.6
-0.4
-0.2
0
Longitudinal Strain (%)
Tran
sver
se s
trai
n (%
)
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of the curve) is constant up to around 2% longitudinal strain where the strain rate has started to
raise. The noise and linearity after 2% longitudinal strain is believed to be damage progression in
the specimen.
The general trends in the data presented here correspond to what has been presented in literature.
However, constant strain rate could not be obtained, and the average strain rate then depend on the
selected strain interval to calculate the average strain rate. Figure 4.9A illustrates the problem
where the strain rate could be set between 100/s and 250/s. Recently, Fitoussi et al. [57] reported
high strain rate testing of FRP composites with a similar setup as used in this work. The time –
strain plot in Figure 4.9B from Fitoussi et al. [57] shows the same problem with a non-constant
strain rate, and the secant line was an attempt to estimate the strain rate. It indicate the problem of
non-constant strain rate persist in the data reported in literature and that the reported strain rate
may be highly dependent on the experimenters choice of interval for calculation of the strain rate.
A) Strain vs time and strain rate vs. time for a
Eglass/Epoxy 0° specimen tested at 20m/s. B) Time vs. strain plot from [57]
Figure 4.9 Comparison of non-constant strain rate problem with results from Fitoussi et al. [57].
If the simulation application is sensitive to strain rate effects, then the calculation of the strain rate
in the test becomes important factor as this affect the rate sensitive parameters values in the
material models. Further, Figure 4.9A also shows that the elastic modulus and failure stress is
estimated at different strain rates for the same test. Any interpretation of rate sensitivity in the
elastic modulus should be done with great care from this kind of high strain rate testing, as the
modulus is estimated when the specimen is strained at low strain rates.
4.4 Summary
An Eglass/Epoxy, An Eglass/Lpet, and C/PA6 were tested at elevated strain rates. General trends
in data corresponded to results reported in literature. However, the C/PA6 showed very high rate
sensitivity, which is doubtful. It was not possible to test 90° specimen at higher strain rate since
preloading in the slack adapter during acceleration caused the specimen to fail before a high strain
rate was reached. It was not possible to reach a constant strain rate in any of the test. In chapter 2
this was explained as an problem with inertial damping. It was pointed out that the variable strain
rate caused problem in stating the strain rate for the test, and the elastic modulus would be
estimated in a strain regime with low strain rate.
0 50 100 150 200 250 300 350 400 450 5000.5
1
1.5
2
2.5
3
Time (s)
Str
ain
(%
)
0 50 100 150 200 250 300 350 400 450 5000
50
100
150
200
250
Str
ain
rate
(/s)
Strain
Strain rate
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5 Introduction to the Split Hopkinson Pressure Bar
The Split Hopkinson Pressure Bar is based on techniques developed by Bertram Hopkinson in
1914 to study wave propagation in long elastic bars [66]. The technique was later used by Davis
and Hunter [67] and Kolsky et al. [68] to create the Split Hopkinson Pressure Bar (SHPB). The
SHPB relies on a single elastic wave to load the specimen, and avoids the oscillations in load
measurements from the high-speed servo hydraulic test machine. The original SHPB was designed
for compression testing, but was later extended to tension, torsion, multi axial, and 3 point bending
tests. The SHPB method is widely described in the literature [17, 18, 20, 69-72] and Zhang et al.
[73] recently gave a comprehensive review of the SHPB method for linear elastic materials. This
chapter briefly introduces the SHPB method with respect to compression testing of linear elastic
brittle materials.
5.1 The Split Hopkinson Pressure Bar
The SHPB consists of two elastic bars with the test specimen positioned between them [8]. One
bar is called the “Incident bar” and the other bar is called the “Transmitter bar”. A schematic is
shown in Figure 5.1 and an photo of the SHPB at the University of Southampton is shown in
Figure 5.2.
Figure 5.1 Schematic overview of the compressive Split Hopkinson Pressure
Bar[69]
A test is conducted by firing a striker bar against the Incident bar. The impact between the bars
creates a compressive pulse, named the “Incident wave”, which travels towards the specimen. The
Incident wave reflects and transmits when it reaches the specimen and the reflected part is named
the “ eflected wave”. The wavelength of the Incident pulse is twice the length of the Striker bar
[3] and much longer than the specimen, such that the Incident wave alone fails the specimen in
compression.
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Figure 5.2 Example of a Split Hopkinson Pressure Bar at the University of Southampton
The transmitted wave propagates through the specimen towards the Transmitter bar and reflects
and transmits again at the specimen/Transmitter bar interface. The transmitted part sets the
Transmitter bar in motion at the interface and the resulting wave travelling in the Transmitter bar
is called the “Transmitted wave”. The reflected part propagates back through the specimen
towards the Incident bar. At the specimen/Incident bar interface, the wave in the specimen reflects
and transmits again and the wave is trapped in the specimen overlapping itself, while the
reflections gradually loses more and more amplitude. At a certain time the wave has overlapped
itself so many times that a state of constant deformation and stress equilibrium is reached and
ideally the specimen first fracture after this state is reached.
The Incident bar, the Transmitter bar, and the Striker bar, are normally made from the same
material and with the same cross sectional area. The striker bar is normally propelled against the
Incident bar with a gas gun driven by either compressed air or a compressed inert gas.
Strain gages are placed on the bars to measure the waves and the strain gages are connected to
high-speed strain gage amplifiers, which condition and amplify the signal from the strain gages to
a high-speed Data acquisition card.
The Incident, the Reflected, and Transmitted wave are measured and the signals are conditioned
and amplified with high-speed strain gage amplifiers and collected with a high-speed data
acquisition board.
Figure 5.3A shows the waves generated in an SHPB with direct impact between the Striker bar
and the Incident bar. According to classic wave mechanics, the direct impact between the Striker
bar and the Incident bar creates a square wave as seen in Figure 5.3A with ripples due to
dispersion [3]. The form of the reflected and transmitted wave is altered by the specimen response.
Figure 5.3B shows an example of pulse shaping where a copper disc is placed between the striker
bar and the Incident bar. The copper disc deforms plastically and creates a ramp Incident wave,
which is beneficial for testing linear elastic materials and soft materials [69, 74]. On the Incident
bar, the Incident and Reflected wave are recorded with the same strain gage. The distance between
the strain gage and the specimen must be at least twice the length of the Striker bar to avoid
overlap of the waves, since the wavelength of the Incident wave is twice the striker bar length in
the case of direct impact [3]. If pulse shaping is applied, the pulse duration is extended to more
than twice the striker bar length and the gage position should be adjusted accordingly [75].
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A) Waves from a standard SHPB with direct impact
between the striker bar and the Incident bar
B) Waves from a SHPB with pulse shaping. In this
case a copper disc was placed between the striker bar
and the Incident bar
Figure 5.3 Examples of stress waves generated in the SHPB
The amplitude of the Incident wave is the particle velocity the wave carries as given by
𝜎 = 𝜌𝐶𝑈𝑝 ⟺
𝜀 =𝑈𝑝
𝐶
(5.1)
σ is stress amplitude, ρ the density of the bar, Up the particle velocity and ε is the strain in the bar.
The total displacement of a material point due to a pass of the wave is obtained by integration of
the wave over its total duration. A low deformation rate requires an Incident wave with low
amplitude and long duration tI to load the specimen to failure. The wavelength is calculated from
the duration and the wave velocity CI in the Incident bar as
𝐿𝑤𝐼 = 𝑡𝐼𝐶𝐼 (5.2)
Most metallic materials have a wave velocity of around 5000m/s and long wave duration easily
requires a very long Striker bar and Incident bar. The gas gun will also have to size with the
Striker bar and in the end there is a practical limit to how low strain rates can be achieved in the
SHPB.
The stress strain curve of the material is obtained by a wave mechanics analysis of the recorded
waves, and the classical Kolsky analysis [68] requires a set of assumptions [68, 71]:
1D Dispersion free wave motion in the bars
The bars remain elastic at all times,
The bar/specimen interfaces remains flat and parallel throughout the entire experiment.
The bars have the same elastic wave speed and diameter.
The specimen is in stress equilibrium.
Strain gages are placed at the Incident and Transmitter bar at the same distances from
the specimen.
Specimen has lower mechanical impedance than the bars.
Figure 5.4 shows the specimen positioned between the Incident and Transmitter bar with the
respective interfaces defined as (a) and (b). A, E and ρ is the cross sectional area, elastic modulus ρ
0 200 400 600 800-1
-0.5
0
0.5
1
Time (s)
Ou
tpu
t vo
ltag
e (V
)
Incident bar
Transmitter bar
Incident waveTransmitted wave
Reflected wave
0 100 200 300 400 500 600-4
-3
-2
-1
0
1
2
3
4
Time (s)
Ou
tpu
t V
olt
age
(V)
Incident bar
Transmitter bar
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density respectively. Subscript I, S and T indicates the Incident bar, specimen and Transmitter bar
respectively. The positive direction is marked (x).
Figure 5.4 Schematic of the specimen situated between
the Incident and Transmitter bars
Ls is the specimen length, and Va and Vb are the interface velocities. The following derivation
follows Kolsky [68] and Meyer et al. [3]. An alternative way to derive the formulas is found in
[71]. In the equations the waves are named as
Incident wave εI and σI
Reflected wave εR and σR
Transmitted wave εT and σT
According to D’Alembert’s solution in equation (1.5), the velocity at a material point affected by
traveling waves is the sum of the positive and negative traveling strain waves multiplied by the
elastic wave velocity of the material. The incident wave reflects at interface (a), and during the
entire test the velocity of interface (a) is calculated from
𝑉(𝑎, 𝑡) = 𝐶𝐼(𝜀𝐼(𝑡) − 𝜀𝑅(𝑡)) (5.3)
For the Transmitter bar the velocity of interface (b) becomes
𝑉(𝑏, 𝑡) = 𝐶𝑇𝜀𝑇(𝑡) (5.4)
The specimen undergoes deformation due to the velocity gradient over the specimen from V(a,t)
and V(b,t), and the technical strain rate becomes
𝑑𝜀
𝑑𝑡= 𝜀̇(𝑡) =
𝑉(𝑎, 𝑡) − 𝑉(𝑏, 𝑡)
𝐿𝑠
(5.5)
Inserting equations (5.3) and (5.4) into (5.5) yields
𝜀̇(𝑡) =1
𝐿𝑠
(𝐶𝐼(𝜀𝐼(𝑡) − 𝜀𝑅(𝑡)) − 𝐶2𝜀𝑇(𝑡)) (5.6)
The strain at time t is found by integrating the strain rate from zero to time t as
𝜀(𝑡) =1
𝐿𝑠
∫ 𝐶1(𝜀𝐼(𝜏) − 𝜀𝑅(𝜏)) − 𝐶2𝜀𝑇(𝜏)𝑡
0
𝑑𝜏 (5.7)
Linear elasticity and D’Alembert’s solution give the forces specimen/bars interfaces
AS, S, S
a b
AI, I, I AT, T, T
a b
s
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𝑃(𝑎, 𝑡) = 𝐴𝐼𝐸𝐼(𝜀𝐼(𝑡) + 𝜀𝑅(𝑡)) 𝑃(𝑏, 𝑡) = 𝐴𝑇𝐸𝑇(𝜀𝑇(𝑡))
(5.8)
The interface stresses then become
𝜎(𝑎, 𝑡) =𝐴𝐼𝐸𝐼
𝐴𝑠
(𝜀𝐼(𝑡) + 𝜀𝑅(𝑡))
𝜎(𝑏, 𝑡) =𝐴𝑇𝐸𝑇
𝐴𝑠
(𝜀𝑇(𝑡))
(5.9)
The stress in the specimen is the average of the interface stresses
𝜎𝑠(𝑡) = 𝐴𝐼𝐸𝐼(𝜀𝐼(𝑡) + 𝜀𝑅(𝑡)) + 𝐴𝑇𝐸𝑇(𝜀𝑇(𝑡))
2𝐴𝑠
(5.10)
Equations (5.7) and (5.10) form the solution for calculating average stress and strain in the
specimen without assumption of equal bars and stress equilibrium in the specimen. Kolsky
discovered the technique of placing the strain gages on the Incident bar and Transmitter bar at the
same distances from the specimen on the Incident and Transmitter bar. Then the Reflected and
Transmitted wave would arrive at the same time at their respective strain gages, which time-
synchronized them. Then Kolsky assumed stress equilibrium in the specimen such that forces
P(a,t) and P(b,t) on each side of the specimen were equal.
𝑃(𝑎, 𝑡) = 𝑃(𝑏, 𝑡) (5.11)
Inserting equation (5.8) into (5.11) yields
𝐴𝐼𝐸𝐼(𝜀𝐼(𝑡) + 𝜀𝑅(𝑡)) = 𝐴𝑇𝐸𝑇(𝜀𝑇(𝑡))
𝜀𝐼(𝑡) + 𝜀𝑅(t) = K𝜀𝑇(𝑡)
(5.12)
with K given as
𝐾 =𝐴𝑇𝐸𝑇
𝐴𝐼𝐸𝐼
(5.13)
If the Incident bar and Transmitter bar are made from the same material and have the same
diameter, then K=1and equation (5.12) reduces to
𝜀𝐼(𝑡) + 𝜀𝑅(𝑡) = 𝜀𝑇(𝑡) (5.14)
Further, the wave velocities in the Incident and Transmitter bars are equal as
𝐶𝐼 = 𝐶𝑇 (5.15)
Inserting equation (5.14) and (5.15) into (5.7) and eliminating the Incident stress, the strain rate
calculation reduces to
𝜀̇(𝑡) = −2𝐶𝐼
𝐿𝑠
𝜀𝑅(𝑡) (5.16)
The strain history is obtained by integration of equation (5.16)
𝜀(𝑡) = −2𝐶𝐼
𝐿𝑠
∫ 𝜀𝑅(𝑡)𝑡
0
𝑑𝑡 (5.17)
Inserting (5.14) and (5.15) into equation (5.10) and eliminating the Incident stress, the specimen
stress calculation reduces to
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𝜎(𝑡) = 𝐴𝐼𝐸𝐼
𝐴𝑠
𝜀𝑇(𝑡) (5.18)
In this case, the specimen stress only depends on the Transmitted wave, while the strain only
depends on the reflected wave. The stress and strain calculations are decoupled from each other
and the time-synchronization is done by the positioning of the strain gages. Alternatively, the
Reflected wave or the Transmitted wave can be eliminated from equation (5.14) instead of the
Incident wave. This forms two new sets of equations, which depend on different waves. The
original olsky formulas are named “One wave ” in Table 5.1 while the two alternative
formulations are named “One wave ” and “Two waves”. “One wave ” only depends on the
Incident and Transmitted waves, which allows alternative arrangements of the strain gage for
increased measurement durations [76]. The “Two waves” formulation uses two waves for
calculation of the stress history, hence its name “Two waves”. The “Three waves” analysis uses all
three waves and does not assume stress equilibrium, but assumes equal bars. If the waves are not
time-synchronized by the gage positioning then they are time-synchronized manually before
applying the equations in Table 5.1. The “Foot shifting” method is when the Transmitter wave is
shifted to rise at the same time as the Incident and Reflected wave. This manual synchronization is
easier to perform, but the method introduces a higher degree of error in the calculations [77]. The
“modern” method refers to the case where no assumptions of equal bars and stress equilibrium are
made in the calculations. The stress and velocity at the ends of the specimen are calculated
separately and used to calculate average stress and strain. The Incident and Transmitter bars can be
of different materials and with different diameters, which is preferable for testing, for example,
soft materials [69].
Table 5.1 Formulas to post process SHPB data – Table partly adopted from Zhang et al. [73]
Name Stress history σ(t) Strain history ε(t) Waves used Assumptions
𝑪𝑰 = 𝑪𝑻 𝜺𝑰 + 𝜺𝑹 = 𝜺𝑻
One wave 1 𝜎(𝑡) = 𝐴𝐼𝐸𝐼
𝐴𝑠
𝜀𝑇(𝑡) 𝜀(𝑡) = −2𝐶𝐼
𝐿𝑠
∫ 𝜀𝑅(𝑡)𝑡
0
𝑑𝑡 𝜀𝑅, 𝜀𝑇 x x
One wave 2 𝜎(𝑡) = 𝐴𝐼𝐸𝐼
𝐴𝑠
𝜀𝑇(𝑡) 𝜀(𝑡) =
2𝐶𝐼
𝐿𝑠
∫ 𝜀𝐼(𝜏) − 𝜀𝑇(𝜏)𝑡
0
𝑑𝜏
𝜀𝐼, 𝜀𝑇 x x
Two waves 𝜎(𝑡) = 𝐴𝐼𝐸𝐼
𝐴𝑠
𝜀𝐼(𝑡) + 𝜀𝑅(𝑡) 𝜀(𝑡) = −2𝐶𝐼
𝐿𝑠
∫ 𝜀𝑅(𝑡)𝑡
0
𝑑𝑡 𝜀𝐼, 𝜀𝑅 x x
Three waves 𝜎(𝑡) = 𝐴𝐼𝐸𝐼
2𝐴(𝜀𝐼(𝑡) + 𝜀𝑅(𝑡) + 𝜀𝑇(𝑡)) 𝜀(𝑡) =
𝐶𝐼
𝐿𝑠
∫ (𝜀𝐼(𝜏) − 𝜀𝑅(𝜏) − 𝜀𝑇(𝜏))𝑡
0
𝑑𝜏 𝜀𝐼, 𝜀𝑅 , 𝜀𝑇 x
Foot shifting 𝜎(𝑡) = 𝐴𝐼𝐸𝐼
2𝐴(𝜀𝐼(𝑡) + 𝜀𝑅(𝑡) + 𝜀𝑇(𝑡 + 𝑡0)) 𝜀(𝑡) =
𝐶𝐼
𝐿𝑠
∫ (𝜀𝐼(𝜏) − 𝜀𝑅(𝜏) − 𝜀𝑇(𝜏 + 𝑡0))𝑡
0
𝑑𝜏 𝜀𝐼, 𝜀𝑅 , 𝜀𝑇 x
Modern 𝜎(𝑎, 𝑡) =
𝐴𝐼𝐸𝐼
𝐴𝑠
(𝜀𝐼(𝑡) + 𝜀𝑅(𝑡))
𝜎(𝑏, 𝑡) =𝐴𝑇𝐸𝑇
𝐴𝑠
(𝜀𝑇(𝑡))
𝜀(𝑡) =1
𝐿𝑠
∫ 𝐶𝐼(𝜀𝐼(𝜏) − 𝜀𝑅(𝜏)) − 𝐶𝑇𝜀𝑇(𝜏)𝑡
0
𝑑𝜏 𝜀𝐼, 𝜀𝑅 , 𝜀𝑇
Hybrid analysis One of the above Direct measurement (DIC etc.) 𝜀𝐼, 𝜀𝑅 , 𝜀𝑇 ( x ) ( x )
Figure 5.5 shows the schematic difference between the “ lassic SHP ” and the “ odern SHP ”.
In the classis set-up (Kolsky), the Incident bar and the Transmitter bar are equal and the strain
gages are placed at equal distance from the specimen. In the modern SHPB, the bars can be of
different materials and different diameters and the strain gages can be placed at different distances
from the specimen. The Modern SHPB requires manual time-synchronization and the calculation
of separate stress and velocities on each side of the specimen.
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Figure 5.5 The classic and modern SHPB setup. Subscript I and T in the modern SHPB
indicate the individual properties of the bars.
5.2 Pulseshaping
If the Striker bar and Incident bar have completely flat ends a nearly square is created upon
impact. However, shaped incident waves with increased rising times facilitate stress equilibrium
and constant strain rate [18, 20, 69, 74].
Christensen et al. [78] introduced pulse shaping to the compressive SHPB by using a conical
striker bar with examples given in Figure 5.6. Recently, Baranowski et al. [79] performed a
numerical study of the conical striker to optimise the Striker shape to different Incident wave
shapes. Ellewood et al. [80] added an extra Incident bar and positioned a specimen similar to the
test specimen between the two Incident bars. The square Incident waves travelled down the first
Incident bar and the extra specimen would then smooth the square wave into the second Incident
bar as a shaped Incident wave with a long rise time.
A) Conical striker [18] B) Incident waves produced by varying the area ratio
between the cylinder and cone [18, 78]
Figure 5.6 Pulse shaping technique – Conical striker bar
Nasser et al. [74] modified the method by Ellwood et al. [80] by removing the extra Incident bar
and exchanged the extra specimen with a soft elasto plastic material such as copper. The Striker
bar would then hit the copper disc directly, which worked as pulse shaper. By varying the diameter
and thickness of the copper disc, Nasser et al. were able to obtain different Incident wave shapes.
Nasser et al. did also describe the deformation process analytically and the description was
extended and improved by Frew et al. [20, 21]. The analytical description allowed the
Incident bar A Transmitter bar A Specimen S S S
Striker bar
Transmitter bar AT T T Specimen S S S
Incident bar AI I I
I T
Striker bar
lassic SHP
odern SHP
Strain gage Strain gage
Strain gage Strain gage
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experimenter to predict the shape of the Incident wave from the pulse shaper dimensions and the
Striker bar length and velocity. The method was later developed into dual pulse shaping with two
pulse shapers on top of each other to create a bi-linear Incident wave [18]. Gerlach et al. [81] have
recently presented a new method of pulse shaping where an extra Incident bar with grooves creates
a set of impedance mismatched, which the incident wave has to pass. The mismatch creates a
series of reflections that smooth the wave.
A) Schematics of a single and dual pulse shaper
setup
B) Example of the copper disc pulse shaper placed at
the Incident bar
Figure 5.7 Pulse shaping method with a cushion material.
The method with a copper disc as cushion material between the striker bar and the Incident bar has
been widely used in the literature [20, 21, 69, 70, 74, 75, 82-84] and is employed in this work. The
next section describes the calibration and utilization of the method.
5.2.1 Copper disc pulse shaper
This section describes the analytical model developed by Nasser et al. and Frew et al. [20, 74] and
the implementation of the method. The assumptions of the model are
The striker bar and Incident bar have equal diameters
The pulse shaper does not deform plastically to a larger diameter than the Incident bar
Figure 5.8 Schematic of the copper disc placed at the Incident bar with the
interface velocities VPS1 and VPS2.
The copper disc has initial thickness h0p, and cross sectional area Ap. The striker bar has density
ρST, elastic wave speed CST, length LST, and impact velocity V0. Upon impact the copper disc
deforms plastically while a compressive wave travels back through Striker bar. The relationship
between time t and strain ε in the copper disc is given by
Striker bar Incident bar
PS
PS
opper disc
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𝑡 =
ℎ0𝑝
𝑉0
∫1
[1 − 𝐾′ (1
𝜌𝐼𝐶𝐼+
1𝜌𝑆𝑇𝐶𝑆𝑇
)𝑓(𝑥)1 − 𝑥
]
𝜀
0
𝑑𝑥, 0 ≤ 𝑡 < 𝑇 (5.19)
f(x) is a function for the copper disc that gives the stress as function of strain. K is defined as
𝐾 =𝜎0𝐴0𝑝
𝑉0𝐴𝑆𝑇
(5.20)
σo is a constant. The time T to traverse the specimen is calculated as
𝑇 =𝐿𝑆𝑇
𝐶𝑆𝑇
(5.21)
The deformation of the copper disc may continue for several round trips of the wave in the striker
bar and the time as function of strain is given by
𝑡 = 𝑛𝑇 +
ℎ0
𝑉0
∫1
[1 − 𝐾 (1
𝜌𝑇𝐶𝑇+
1𝜌𝑆𝑇𝐶𝑆𝑇
)𝑓(𝑥)1 − 𝑥
] −2𝐾
𝜌𝑆𝑇𝐶𝑆𝑇∑
𝑓(𝜀(𝑡 − 𝑘𝑇))1 − 𝜀(𝑡 − 𝑘𝑇)
𝑛𝑘=1
𝜀
𝜀𝑛
𝑑𝑥,
𝑛𝑇 ≤ 𝑡 < (𝑛 + 1)𝑇
(5.22)
𝜀𝑛 is the strain at t = nT. The pulse shaper compresses when VPS1> VPS2. When VPS1<VPS2
equations (5.19) and (5.22) are no longer valid and the pulse shaper is assumed to be elastic
unloading. The interface velocities VPS1 and VPS2 are calculated from equations (24a) and (24b) in
[21]. Frew et al. suggested a resistance function for the copper disc as
𝜎𝑝 = 𝜎0𝜀𝑝
𝑛
1 − 𝜀𝑝𝑚
(5.23)
The function is calibrated by an end-point method. A series of tests were conducted and after each
test, the maximum strain was derived from the final thickness of the copper disc. The associated
maximum stress (𝜎𝑝) was found from the generated Incident wave as
𝜎𝑝 =𝐸𝐼𝐴𝐼
𝐴𝑃
(1 − 𝜀𝑝) ∙ max (𝜀𝐼(𝑡)) (5.24)
The acquired data were plotted in a stress strain plot and equation (5.23) was fitted to the data
point in a least square sense to determine n, m, and σ0.
A soft grade stock copper was available in sheets of thickness of 0.5, 1, 1.5, 2 and 3mm. Round
discs were created in various diameters d0 between 3 and 15 mm with a punch machine available
in the tool shop. This is laboratory practice to create copper disc pulse shapers as noted by Nishida
et al. [85].
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A) Copper sheets after punching B) Different size pulse shapers
between 3 and 15mm after test
C) Striker bars in lengths of 50mm,
100mm, 150mm and 200mm made
from stainless steel. The Incident bar
was made from the same material.
Figure 5.9 Copper discs and available strikers. The Teflon coating shown together with the Striker bars, was
used for lubricating the barrel of the gas gun and is. Lubrication, as WD40, is not preferred as the tight
tolerances in the barrel causes the oil to pile up in front of the Striker bar and disturbs the impact.
In total, twenty tree tests were used to calibrate the resistance function in equation (5.23). LST, d0,
h0p and V0 were varied between the tests to spread the data point along the line. Figure 5.10A gives
the plot with the fitted function along with the estimated parameters of equation (5.23). Figure
5.10B shows a schematic overview of pulse created with a copper disc pulse shaper [86]. Part 1 is
the elastic response of the pulse shaper. Part 2 is the plastic response and part 3 is the rigid
response where the pulse shaper is maximum compressed. Part 4 is elastic unloading of the pulse
shaper. An example of a measured Incident wave and a simulated wave is shown in Figure 5.10C.
The measured wave does not have the initial elastic response as the simulated curve, which
indicates that some form of damping was present at the impact.
A) Fitted resistance function
σ0=608.3, m=4.236, n=0.2387.
B) Schematic graph of a shaped
incident wave using a copper disc.
Adopted from [86]
C) An example of a simulated and
the corresponding measure
Incident wave.
Figure 5.10 Pulse shaping with a copper disc
There are four parameters that control the shape of the Incident wave: V0 = striker bar impact
velocity; LST = the striker bar length; h0 = the pulse shaper thickness; and d0 = the pulse shaper
diameter. Figure 5.11 shows a parametric study the shaped incident wave was simulated for a set
of parameters.
Increased impact velocity increases the stress imposed on the pulse shaper. If in the impact
velocity is high enough, the pulse shaper compresses enough to go into a strain region (εtech > 0.6 )
with a very steep increase of stress as seen for V0 = 25m/s. An increasing striker bar length
extends the pulse duration and forces the pulse shaper to higher strains. If the striker bar is long
enough, the pulse shaper compresses to strains beyond 0.6 and the pulse becomes highly
nonlinear. This is seen for LST = 300mm.
An increasing diameter causes the initial elastic response to be higher, as the pulse shaper exerts a
0 0.2 0.4 0.6 0.80
500
1000
1500
Engineering strain
Stre
ss (
MP
a)
Calibration data for Pulseshaper UoS
Datapoints
Fit
0 50 100 150 200 2500
50
100
150
200
Time (s)
Stre
ss (M
Pa)
Measured pulse
Simulated pulse
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higher force on the Incident bar before the pulse shaper yields. If the striker bar length and velocity
are kept constant, an increased diameter decreases the amount of plastic strain. Thus, the response
grows in amplitude with increased diameter but moves backwards on the pulse shaper stress strain
curve in Figure 5.10B.
An increased thickness lowers the amplitude of the generated pulse but extends the pulse duration.
The increased thickness leads to a softer response, which gives a lower deceleration of the striker
bar and thus a longer time in contact with the striker bar. The higher thickness also means that the
induced strain is less and the pulse shaper is not loaded so far up its stress strain curve and keeps
the amplitude low on the generated wave.
Figure 5.11 Parameter study of the four parameters. The basic set of parameters are the striker bar velocity
V0, the striker bar length, the pulse shaper diameter and the pulse shaper thickness.
Rising Incident waves with approximately constant and different stress rate can be generated by
varying the four parameters. The copper disc method has limitations with respect to generating a
monotonic rising pulse, required for a smooth loading of the specimen. If the parameters are
misadjusted, the generated stress wave increases in rate at the last part. This happens when the
pulse shaper is deformed into a strain range with high effective modulus taken as the slope of the
stress strain curve in Figure 5.10B. This is seen in Figure 5.12 for copper with d0 = 6mm compared
with d0 = 7.4 mm which has a more linear stress rate. As illustrated in Figure 5.11, the increased
diameter decreases the maximum strain in the pulse shaper, but increases the initial elastic
response as seen here. The straightforward way to increase the stress rate would be to increase the
diameter and adjust the other parameters accordingly to obtain a linear response of required
duration. However, the model presented by Frew et al. [21, 74] does not account for friction. For
the combination of the bars and copper material used in this study the limiting diameter was about
11-12mm. At higher diameters, severe ripples were created in the Incident wave. This causes an
oscillating strain rate and addition of high frequency content to the wave.
0 50 100 150 200 2500
50
100
150
200
250
300
350
400
450
Striker bar velocity
Time (s)
Incid
en
t b
ar
str
ess (
MP
a)
V0: 2 m/s
V0: 10 m/s
V0: 15 m/s
V0: 25 m/s
0 50 100 150 200 2500
20
40
60
80
100
120
140
Striker bar length
Time (s)
Incid
en
t b
ar
str
ess (
MP
a)
LST
: 50 mm
LST
: 100 mm
LST
: 200 mm
LST
: 300 mm
0 50 100 150 200 2500
20
40
60
80
100
120
140
160
Diameter
Time (s)
Incid
en
t b
ar
str
ess (
MP
a)
d0: 2 mm
d0: 6 mm
d0: 10 mm
d0: 15 mm
0 50 100 150 200 250 300 3500
20
40
60
80
100
120
140
160
180
Thickness
Time (s)
Incid
en
t b
ar
str
ess (
MP
a)
h0: 0.5 mm
h0: 1 mm
h0: 2 mm
h0: 4 mm
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Figure 5.12 Various diameters (D) of 2mm thick copper disc
tested under similar conditions.
A series of tests were carried out to determine the controlling parameters of the ripple problem.
Figure 5.13A shows that at 1mm thickness the ripple problem only persists at diameters above
10mm. Figure 5.13B shows that increased impact velocity does not provoke the ripples to occur.
In this case, the diameter and thickness were fixed and the impact velocity varied. Figure 5.13C
shows that the increased thickness does not damp the ripple problem when the diameter was
increased up to the critical dimensions. The repeated test in Figure 5.13C shows that the ripple
problem persists and is not random.
A) 1 mm thick copper discs B) 2mm thick copper discs C) 3mm thick copper discs
Figure 5.13 Investigation of “ripples”. Pulse shapers were tested under various conditions to narrow down
the controlling parameters of ripples.
The results in Figure 5.13 show that the diameters is the controlling parameter. Several lubricants,
such as grease, MoS2, and thin plastic foil were tried, but did not damp the ripples in a
reproducible way. The compression of the pulse shaper resembles a compression test of ductile
materials in the SHPB so inspiration from these tests was searched for in the literature. An
accepted design to balance inertia problem is a ratio h0/d0 = 0.47 [87] for a material with Poisson
ratio of 0.3. The 11.42mm diameters discs of 3mm thickness had an h0/d0 ratio of 0.263. If friction
were of concern then the h0/d0 ratio should be between 1.5 and 2 to minimize the effect of friction
0 50 100 150 200 250 300 3500
50
100
150
200
250
Time (s)
Stre
ss (M
Pa)
D = 6mm, V0 = 13.60m/s, Ls = 200mm
D = 7.4mm, V0 = 13.29m/s, Ls = 200mm
D = 9 mm, V0 = 15.13m/s, Ls = 200mm
D = 11.4mm, V0 = 15.68m/s, Ls = 200mm
D = 14.5mm, V0 = 14.60m/s, Ls = 200mm
0 50 100 150 200 250 300 3500
50
100
150
200
250
Time (s)
Stre
ss (M
Pa)
D = 5mm, V0 = 11.31m/s, Ls = 200mm
D = 5mm, V0 = 16.50m/s, Ls = 100mm
D = 10mm, V0 = 14.83m/s, Ls = 200mm
D = 15mm, V0 = 12.05m/s, Ls = 200mm
0 50 100 150 200 250 300 3500
50
100
150
200
250
Time (s)
Stre
ss (M
Pa)
D = 9mm, V0 = 5.43m/s, Ls = 150mm
D = 9mm, V0 =15.75m/s, Ls = 150mm
D = 9mm, V0 = 23.66m/s, Ls = 50mm
0 50 100 150 200 250 300 3500
50
100
150
200
250
Time (s)
Stre
ss (M
Pa)
D = 11.42mm, V0 = 22.51m/s, Ls = 100mm
D = 11.42mm, V0 = 21.98m/s, Ls = 100mm
D = 11.42mm, V0 = 21.73m/s, Ls = 100mm
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on the result [88], but this is with respect to the measured signal and not with respect to ripples in
the signal. An h0/d0 ratio in this range is not feasible for pulse shapers as the generated waves will
have too small amplitude. Naghdabadi et al. [86] performed LS Dyna simulations of copper disc
pulses shapers. Their simulations showed the ripples as function of the diameter, but the authors
did not address the reason for the ripples or mention them as a problem. Figure 5.14 shows some
of the simulations.
Figure 5.14 LS Dyna simulation of pulse shapers.
hp= 1mm, V0 = 16m/s, LST = 400mm [86].
Then the pulse shaper method with a copper disc as cushion material between the striker bar and
the Incident bar is limited by the frictional properties between the pulse shaper and the striker and
Incident bars. The pulse shaper diameter controls the initiation of the ripple problem. For the
specific combination materials used in this study the maximum diameter was around 10-11mm
before the Incident wave got to large ripples.
5.3 Calibration of the SHPB setup
Post processing of the data acquired from an SHPB test requires the elastic modulus and the
density to calculate the elastic wave velocity. Any error in these values propagates directly into the
calculated values of stress and strain. Lifshitz et al. [89] examined the required accuracy of the
elastic wave velocity of the incident and Transmitter bar and found that just a 1% deviation caused
errors in the post-processed results as shown in Figure 5.15. They advised against using tabular
values for density and the elastic modulus, but instead do a calibration.
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Figure 5.15 Effect of the accuracy of the elastic wave velocity C0
on the stress strain curve [89].
The density ρ can be determined by measuring the weight and the dimensions of the bar. The
elastic modulus E is estimated either from a tensile test, or by measuring the elastic wave velocity
and applying 𝐶 = √𝐸
𝜌. The SHPB test rig itself can conveniently be used to measure the wave
velocity. In the Incident bar, the strain gage is positioned at distance LSGI. Figure 5.16A shows a
typical recording from the strain gage with the Incident and Reflected waves in the signal. The
time Δt is the time it takes the Incident wave to propagate to the specimen and the eflected wave
to come back to the strain gage again. The elastic wave velocity can then be estimated as [18]:
𝐶𝐵 =2𝐿𝑆𝐺𝐼
∆𝑡 (5.25)
Δt is normally measured manually by the e perimenter by determining when the Incident and
Reflected pulses rise as indicated in Figure 5.16A. The estimate of Δt relies on the e perimenter’s
ability to judge where the signals start and end and is a source of error. In Figure 5.16B the effect
of judging the time μs/ μs wrong is shown as a function of the strain gage position and this
should be compared to the time scale in Figure 5.16A. The estimation of the elastic wave velocity
is then strongly dependent on the noise level in the signal and the e perimenter’s ability to judge
where the wave rises.
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A) Example of Incident and reflected pulse in the
Incident bar for gage position of l=1m. Δt is the time
it takes the wave to travel to the end of the bar and
back to the strain gage.
B) Error in estimated C0 if time is estimated 1 μs/5 μs
wrong for a real elastic wave velocity of 5000m/s.
Figure 5.16The incident and reflected wave in the Incident bar and the error in estimating the elastic wave
velocity
5.3.1 Alternative method for estimating C0
To remove the dependency on the e perimenter’s ability to judge the start of the incident and
reflected wave, an alternative method can be used to estimate the wave velocity. The method
exploits the longitudinal resonance phenomena of a bar [16].
Assume the Incident wave can move freely without contact to the specimen or the Transmitter bar.
The impact of the striker bar creates a compressive wave traveling in the positive direction x of the
Incident bar as indicated in Figure 5.17. At the free end of the bar, the wave reflects completely
and returns as a tensile wave traveling in the negative direction. At left the end, it reflects and
travels in the positive direction as a compressive wave again. This process continues until external
and internal friction has reduced the amplitude of the wave to zero, and the bar is at rest.
Figure 5.17 Schematic of the wave traveling inside the Incident bar. The
striker bar impact creates a compressive wave, which travels in the positive
direction. As the wave reflects at the other end it return as tensile wave
traveling in the negative direction.
The strain gage at the Incident bar records both the compressive and tensile waves, and the time it
takes the wave to reach back to the strain gage as a compressive wave after last time it passed as a
compressive wave is marked with a red arrow in Figure 5.18A. The red arrow represents the
0 0.5 1 1.5 20
0.5
1
1.5
2
Gage position l (m)
CE
stim
ate
/C0 (%
)
-1s error in time estimation
-5s error in time estimation
Incident bar
L
Striker
L
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period T, which equals the time it takes the wave to travel 2𝐿𝐼 . The inverse of the period gives the
frequency so the frequency of the signal in Figure 5.18A yields
𝑓𝐼 =𝐶𝐼
2𝐿𝐼
(5.26)
The frequency is found by a Fourier transformation of the strain gage signal. The frequency
corresponding to the wave velocity which will have the highest energy in the frequency domain.
The strain gage signal in Figure 5.18A was obtained on a bar of stainless steel with a measured
density of 7741kg/m3 and a length 2.001m. The wave speed should be around 5000m/s such that
the frequency should be around 5000𝑚/𝑠
2.001𝑚∗2= 1250𝐻𝑧. Figure 5.18B shows the frequency spectrum
of the signal in Figure 5.18A.
A) Period of the signal marked with an red arrow B) Frequency spectrum of signal shown in (A)
Figure 5.18 A signal from the Incident bar along with its representation in the frequency domain.
There is a clear peak at 1242Hz, which corresponds to a wave velocity of 4970 m/s. There are
peaks at frequencies that are multiples of 1242Hz but these are harmonics and should be ignored.
Thus, an algorithm can be set up that transfers the strain gage signal from the time domain to
frequency domain, and then performs a search for the highest peak near the expected frequency of
the wave velocity. The expected frequency is calculated from tabular values of the elastic
modulus, the density, and the measured bar length.
The method is proved to be robust because the dependency on the e perimenter’s judgement of the
signal is removed. Further, the method is also immune to noise as the energy of the noise will be
less than that of the wave velocity frequency and be of higher frequency. The strain gage signal is
discretely sampled in time at sampling frequency fs. The Fourier transformation then becomes a
discrete sampling of the underlying analogue frequency spectra. If the signal length in data points
is N then the resolution of the frequency spectra yields
∆𝑓 =𝑓𝑠
𝑁 (5.27)
The sampled length N should be maximised to increase the accuracy of the method. In this work a
resolution was 1Hz. Instead of recording longer signals, the signal can be padded with zeroes to
increase the resolution. For this work, an Incident bar and a Transmitter bar were calibrated using
0 0.5 1 1.5 2 2.5 3
-4
-3
-2
-1
0
1
2
3
4
Time (ms)
Ou
tpu
t vo
ltag
e (V
)
0 2000 4000 6000 8000 100000
0.5
1
1.5
X: 1242
Y: 1.425
Frequency (Hz)
Am
plit
ud
e
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the proposed method. The density was determined by measuring the dimensions and the weight
three times and the average values were used. The wave velocity was calibrated through 10 tests to
estimate the stability of the method. Table 5.2 shows the results. The wave velocities were
estimated with a 95% confidence interval of ±2m/s, which is a relative error of ±0.04%. Thus, the
method outperforms the manual method in precision and accuracy. The experimenter will have to
select the correct period with a precision better than 0.15µs for strain gage position of 1m to reach
a relative error of 0.04%.
Table 5.2 Calibration values for the incident and Transmitter bar.
Parameter Incident bar Transmitter bar
Material Stainless steel Aluminium
Length 2001mm 617mm
Diameter 25.02mm 24.99mm
weight 7.616kg 0.85kg
Calculated density 7741 kg/m3 2809kg/m3
Estimated wave velocities
Test 1 4974 m/s 5105 m/s
Test 2 4974 m/s 5111 m/s
Test 3 4974 m/s 5110 m/s
Test 4 4982 m/s 5111 m/s
Test5 4978 m/s 5115 m/s
Test 6 4970 m/s 5107 m/s
Test 7 4970 m/s 5116 m/s
Test 8 4972 m/s 5112 m/s
Test 9 4972 m/s 5111 m/s
Test 10 4976 m/s 5106 m/s
Average 4974 m/s 5110 m/s
95% confidence interval 4972 < CI < 4976 m/s 5108 < CT < 5112 m/s
Elastic modulus 191GPa 73.5GPa
5.4 Time synchronisation
The Incident wave is recorded before the deformation takes place and the reflected and transmitted
wave are recorded after the deformation took place. However, during the deformation, the waves
are present at the specimen at the same time. To perform the post processing of the signals, the
waves must be offset in time to be aligned as they would if they were recorded directly at the
specimen ends. In the classical SHPB, the signals are automatically aligned since there is the same
distance from the specimen to the strain gage on both bars. In a modern SHPB with different strain
gage positions, the signals need to be synchronized after the test. In this work the rise of the
incident wave was chosen as the synchronization point. If the rise of the incident wave is taken as
T0 on a global timescale T, then according to [18] the reflected wave rises at the strain gage at
𝑇𝑅𝑡𝑜𝑡 = 𝑇0 + 𝑇𝑅 = 𝑇0 +2𝐿𝑆𝐺𝐼
𝐶𝐼
(5.28)
The transmitted wave then rises at
𝑇𝑇𝑡𝑜𝑡 = 𝑇0 + 𝑇𝑇 =𝐿𝑆𝐺𝐼
𝐶𝐼
+𝐿𝑆𝐺𝑇
𝐶𝑇
+𝐿𝑠
𝐶𝑠
(5.29)
LSGT is the gage position on the Transmitter bar with respect to the specimen, CT the wave velocity
of the Transmitter bar, Ls the specimen length and Cs the specimen wave velocity.
The reflected wave is synchronized by shifting the time of all data recorded on the Incident bar
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after TRtot as
𝑇𝑅𝑠ℎ𝑖𝑓𝑡𝑒𝑑= 𝑇 − 𝑇𝑅 (5.30)
The transmitted wave is synchronized by shifting the vector of all data by
𝑇𝑇𝑠ℎ𝑖𝑓𝑡𝑒𝑑= 𝑇 − 𝑇𝑇 +
𝐿𝑠
𝐶𝑠
(5.31)
The transmitted wave cannot be shifted 𝐿𝑠
𝐶𝑠 backwards as this travel time through the specimen was
present at the time when the actual deformation took place. However, if extra shifting is done and
TT alone is used for time shifting, the method is called foot shifting and causes higher errors in the
estimated stress-strain curves [77].
Finally, the shift value for the transmitted waves becomes
𝑇𝑇𝑐 = 𝑇𝑇 +𝐿𝑠
𝐶𝑠
=𝐿𝑆𝐺𝐼
𝐶𝐼
+𝐿𝑆𝐺𝑇
𝐶𝑇
(5.32)
The shift value for the reflected wave becomes
𝑇𝑅 =2𝐿𝑆𝐺𝐼
𝐶𝐼
(5.33)
Figure 5.19A shows the shift times in relation to each other and the raw waves. Figure 5.19B
shows the waves after the shifting was performed.
A) Stress reading from the incident and Transmitter
bar. B) Waves after synchronisation.
Figure 5.19 Synchronisation of the Incident, Reflected and Transmitted wave.
The synchronization can only be done successfully if the wave velocities and strain gage position
are measured with high precision. Otherwise, the synchronization leads to erroneous results in the
post processing process.
5.5 Dispersion correction
A stress wave propagating along a slender bar can be expressed as a sum of harmonic waves with
different amplitudes, periods and phases. The SHPB post processing procedures assume simple 1D
0 200 400 600 800 1000-150
-100
-50
0
50
100
150
Time (s)
Stre
ss r
ead
ings
(MP
a)
Incident bar
Transmitter bar
T TTc
T
0 100 200 300 400-150
-100
-50
0
50
100
150
Time (s)
Stre
ss (M
Pa)
Incident wave
Reflected wave
Transmitted wave
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wave theory which states the wave velocity of a harmonic sinusoidal wave in an elastic rod to be
calculated by 𝐶 = √𝐸/𝜌. However, if the theory is expanded to 3D analysis including Poisson
effect, lateral material movement and inertia are also introduced. The general 3D theory for
propagation of sinusoidal longitudinal waves in long slender bars was derived by Pochhammer and
Chree [67, 90] and concluded that the sinusoidal wave velocity depends on
R - The bar radius
Λ - Wave length
C0 - Wave velocity of infinitely long wave – 1D wave velocity.
Ν - Poisson’s ratio of the bar material
The original equations are complex and have an infinitely number of solutions corresponding to
the different vibrational modes of the bar. The first mode (solution), is vibration in the longitudinal
direction of the bar and is the dominating part in an SHPB test [72]. Figure 5.20 shows the exact
solution of the Pochhammer-Chree equation to the three first vibrational modes of a longitudinal
bar of infinite length. C is the wave velocity of a wave with wavelength Λ.
Figure 5.20 Exact Solution of the first 3 vibrational modes to
the Pochhammer-Chree equation for bar material with ν=0.3.
Mode 1 is dominant in SHPB tests [72].
Figure 5.20 shows that wave components with wavelengths less than the bar radius travels 40-50%
slower than waves with long wavelengths. Thus, high frequency content travels slower than low
frequency content, resulting in a dispersion of the wave where the rising and falling edges are
smoothed out and high frequency oscillation is superimposed on the original wave as shown in
Figure 5.21 [90].
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Figure 5.21 Initial sharp wave subjected to dispersion
The wave shape has an impact on the degree of dispersion, since a pulse with a short rise time has
a wide frequency band with a higher amount of high frequency content, while a pulse with a
smooth rising edge has a narrow low frequency band and suffers less from dispersion [91]. This is
seen in Figure 5.22 where the frequency spectra were calculated for a square wave and a ramp
wave recorded at the SHPB used in this work.
A) A square and pulse shaped incident wave B) Frequency content
Figure 5.22 Incident waves and their associated frequency content
According to the Pochhammer-Chree equations [67, 72], there is 3 main problems with dispersion
1. High frequency components travel with a slower propagation velocity than low frequency
components.
2. Uneven distribution of stress across the cross sectional area of the bar as function of 𝑅
Λn
3. High frequency elastic modulus is different from the quasi-static elastic modulus.
Since dispersion corresponds to a phase shift as the wave travels, dispersion is corrected by
performing a Fourier transformation of the wave to the frequency domain and a time shift of the
phase angles for frequency in the wave. When the phase angles have been shifted, the pulse is
shifted back into the time domain by an inverse Fourier transformation [91]. The incident wave is
shifted forward in time as the wave changes after the strain is recorded and before the specimen is
0 100 200 3000
0.2
0.4
0.6
0.8
1
Time (s)
No
rmal
ised
Ou
tpu
t vo
ltag
e (V
/Vm
ax)
Square pulse
Shaped pulse
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
f(kHz)
No
rmal
ised
En
ergy
Square pulse
Shaped pulse
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loaded. The reflected wave and transmitted wave have to be shifted backwards as these waves are
recorded after the specimen is loaded.
Andrew et al. [92] gave a step-by-step overview of the dispersion correction technique:
1. Convert the time-domain signal f(t) into the frequency domain
2. Derive the amplitude ω and phase angle ϕ ω of each frequency component.
3. alculate the phase velocity ω of each frequency component.
4. For each frequency component, calculate the phase shift from the position at which the signal
is recorded to some other point at an axial distance z from the recording position. z is taken
positive in the propagation direction of the wave. The phase shift is ϕ’ ω in radians is given
by
𝜙′(𝜔) = (𝐶0
𝐶(𝜔)− 1)
𝜔𝑧
𝐶0
(5.34)
5. Convert the signal back into the time domain using a corrected phase angle for each frequency
The dispersion correction is necessary if the frequency content in the bar is high. However, at
lower frequencies the dispersion is not distinct. Figure 5.24 shows the phase velocity calculated
from the 1D wave equation and the exact solution to the Pochhammer-Chree equations. As long as
�̅� < 0.1 there is a match between wave equation and the exact theory and thus this can be used as
measure if dispersion correction is required.
Figure 5.23 Variation of the phase velocity as function of the normalised
wave number.
The normalized wavenumber is defined as
�̅� =2𝜋𝑘𝜈𝑓
𝐶0
(5.35)
k is the polar radius of gyration calculated given by
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𝑘 = √𝐽𝑦𝑦
𝐴= √
𝜋𝑟4
2𝜋𝑟2=
𝑟
√2 (5.36)
Figure 5.24 shows the maximum frequency as function of the bar radius for a material with
C0=5000m/s. Thus, for a 25mm bar the maximum frequency content should be less than 92 kHz.
Comparing with Figure 5.22 the shaped Incident wave does not have much energy above 10 kHz
and pulse shaping thus helps to minimize the dispersion effects in the SHPB.
Figure 5.24 Maximum frequency present in the
incident wave before dispersion correction is required.
v = 0.3, C0= 5000m/s.
5.6 Summary
The compressional Split Hopkinson Pressure Bar to test linear elastic materials, were described in
detail. The concept of pulse shaping was presented and particularly the method of utilizing cushion
materials as copper. This method was found to be limited by friction, which created ripples in the
shaped wave. Calibration of the SHPB test rig itself was also described. A new method to
accurately calibrate the wave velocity in the bars was described which minimised the influence
from the experimenter and could calibrate the wave velocity to a higher degree of accuracy than
existing methods. It was emphasized that for a precise time-synchronization great attention has to
be paid to an accurate calibration of the SHPB test rig. Lastly, it was shown that as long pulse
shapers are used, dispersion is minimised, which simplifies the post processing of the recorded
data.
0 20 40 60 80 1000
50
100
150
200
250
300
350
X: 25
Y: 92.66
Bar diameter (mm)
Max
freq
uen
cy (K
hz)
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6 A generalized wave mechanic model of the Split Hopkinson Pressure Bar
6.1 Introduction
The original Kolsky formulas and the alternative derivations for calculating the stress-strain
response all assume equilibrium over the specimen [22][8]. Alternatively, separate stress measures
from the incident and Transmitter bar can be used as presented for the modern SHPB in chapter 5.
However, in each case, the conversion of the data into a stress-strain curve implicitly assumes a
homogenous deformation state of the specimen, and an assessment of when the specimen is in
equilibrium is required. A generalized model is developed in this chapter to perform this analysis
for linear elastic materials.
Ravichandran et al. [22] reported early in 1994 a model based for a linear elastic specimen tested
in a classic SHPB. They calculated how many times the elastic wave had to traverse the specimen
before the specimen had reached a state of equilibrium. Equilibrium was defined as stress
difference between the end to be less than 5% of the mean stress in the specimen and calculated as
𝑅𝐸𝑄 = |𝜎𝑎 − 𝜎𝑏
𝜎𝑎 + 𝜎𝑏
2
| (6.1)
Then the maximum strain rate is calculated from the number of wave transits to equilibrium and an
assumption of constant strain rate. Parry et al. [93] presented a wave mechanics model for the
classic SHPB, and the model described how the stress developed at the specimen/bar interfaces.
They defined an equilibrium criterion as
𝐸𝑄 =𝜎𝑇
𝜎𝐼 + 𝜎𝑅
(6.2)
σI σR and σT are the incident, reflected and transmitted waves respectively and the bars are assumed
to be equal in terms of the mechanical impedance. When the criterion reached unity, the specimen
was in stress equilibrium. They found that stress equilibrium was reached within 3-5 wave transits
dependent on the mechanical impedance of the tested material.
Frew et al. [20, 21] presented two models for calculating the evaluation of strain and stress in
linear elastic material. They first used the original Kolsky formulas with assumption of stress
equilibrium and equal bars to derive a closed form differential solution for the evolution of strain
rate and strain through a test with linear rising pulse as loading. Equation (6.3) gives the solution
for strain rate calculations and equation (6.4) gives the solution for calculating the strain.
𝜀̇(𝑡) =𝐴𝐵𝑀
𝐴𝑠𝐸𝑠
[1 − 𝑒−2𝑡𝑟𝑡𝑜 ] (6.3)
𝜀(𝑡) =𝐴𝐵𝑀
𝐴𝑠𝐸𝑠
{𝑡 −𝑟𝑡0
2[1 − 𝑒
−2𝑡𝑟𝑡𝑜 ]} (6.4)
Equations (6.3) and (6.4) were later used by Pan et al. [70] and Foster et al. [94] to derive closed
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form solutions for the maximum strain rate maintaining the inherent assumption of stress
equilibrium. Table 6.1 compares the maximum strain rate measures and their assumptions.
Table 6.1 Maximum strain rate criteria for linear elastic brittle specimens and equal bars
Reference Limit strain rate criteria Assumption
[22] 𝜀�̇�𝑖𝑚 =𝜀𝑓
𝑛𝑡0
Constant strain rate
[70] 𝜀�̇�𝑖𝑚 =𝜀𝑓
𝜏1 − 𝜉
−𝑟𝑡0
2
Stress equilibrium, linear rising pulse
[94] 𝜀�̇�𝑖𝑚 =
𝜉 − 1
1 + ln(𝜉) − 𝜉
2𝜀𝑓
𝑟𝑡0
Stress equilibrium, linear rising pulse
Parameter Description
n Number of wave transits in the specimens
𝜀𝑓 Failure strain
𝜎𝑓 Failure stress
𝜉 Accepted deviation from equilibrium.
Typically 0.05.
τ Time to constant strain rate
𝑡0 =𝐿𝑠
𝐶𝑠
Propagation time through the specimen for an
elastic wave.
𝑟 =𝜌𝐵𝐶𝐵𝐴𝐵
𝜌0𝐶0𝐴0
Relative impedance between bar and
specimen
Ravichandran et al. [22] found that a linear rising loading pulse requires at least four transits of the
elastic wave through the specimen to occur before the stress difference was below 5% of the mean
stress. This was calculated for a mechanical impedance ratio between the bar and specimen r = 1/8.
For higher ratios more transits were required. The second set of equations Frew et al. [20, 21]
developed is based on 1D wave mechanics and summed up all the wave reflections at the ends of
the specimen as the wave propagated back and forth inside the specimen. The method used by
Frew et al. [21] is in principle equivalent to the wave mechanics method used by Parry et al. [93],
but Parry et al. arranged their formulas so stresses were summed up in the bars and not in the
specimen. Yang et al. [19] used the same method as Frew et al. to derive equations for the stress
equilibrium as defined in equation (6.1). Both step pulses and linear rising pulses were considered
and Yang et al. found that at least six wave-transits were required for a linear rising pulse to get
the stress difference below 5% of the mean stress in the specimen. Then there is not agreement in
literature about the time to stress equilibrium. Table 6.1 shows the same for the criteria for
constant strain rate. Further, all solutions in Table 6.1 also assume either stress equilibrium or
constant strain rate. Foster et al. [94] used the strain rate equation from Frew et al. and found that
stress equilibrium always occurs before constant strain rate, but the conclusion is limited by a
assumption of uniform deformation in the specimen. Equilibrium solutions have also been found
by modelling the specimen as a dynamic spring dashpot system as presented by Song et al. [17],
but without agreeing with the other solutions.
The strain can be hard to measure when soft materials are tested. A high impedance mismatch
between the specimen and Transmitter bar creates very low amplitude strains in the Transmitter
bar, which are difficult to accurately measure. By suitable impedance matching, the signal can be
amplified. For soft material testing, the impedance of the bar(s) must be lowered by changing the
material and/or cross sectional area to match the soft materials to test. Chen et al. [95] changed the
bar material from steel to aluminium and further used a hollow aluminium bar as the Transmitter
bar for testing rubber, but the usage of uneven bars alters the Kolsky formulas and complicates the
data reduction. Designing a SHPB requires the experimenter to decide the specimen cross
sectional area and specimen length from requirements of high enough (constant) strain rate and
homogenous deformation state (stress equilibrium) at failure. This is done from the models of
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stress equilibrium and constant strain rate. However, the presented criteria for stress equilibrium
and constant strain rate are no longer valid as the total impedance of the incident and Transmitter
bars are not equal. The experimenter also has to select an Incident wave to apply to the specimen
that satisfies the requirement of the test. The Incident wave must be shaped to promote stress
equilibrium and constant strain rate.
In this chapter, a model is developed that describes the balancing of specimen cross sectional area
and gage length with bar impedance and the form and size of the incident wave. The model is
formulated and generalized such that restrictions of equal bars are removed. For reference, the
model is named the GW model.
6.2 1D generalised wave mechanic model
The wave mechanics principle presented by Parry et al. [93] and Frew et al. [21] is used here to
develop the GW model. Figure 6.1 defines a number of constants along with interface (a), between
the Incident bar and the specimen, and interface (b), between the specimen and the Transmitter
bar. ‘A’ is the cross sectional area, the elastic modulus and ρ the density.
Figure 6.1 Schematic overview of the specimen and bar
interfaces. The interface between the Incident bar and the
specimen is denoted (a), while the interface between the specimen
and the Transmitter bar is denoted (b). A is the cross sectional
area. E is the elastic modulus, and ρ is the density. x defines the
positive propagation direction of waves in the specimen.
The process starts with the incident wave propagating in the Incident bar towards the specimen.
When the wave reaches the specimen, the wave is partly reflected and transmitted at interface (a).
The transmitted part travels through the specimen, reaches interface (b) where it reflect, and
transmits again. The reflected part from interface (b) propagates back toward interface (a) where it
reflects and transmits again. The following assumptions apply to the wave motion:
1D dispersion free wave motion.
The specimen behaves linear elastic up to fracture (brittle specimen).
Single loading wave from the Incident bar
Specimen remains in perfect contact with the bars during the entire loading duration
(force equilibrium over the individual interfaces.)
Bar ends remains perfectly flat during the entire loading duration.
The bars have higher mechanical impedance than the specimen.
Equation (6.5) describes the 1D dispersion free wave motion [16].
AS, S, S
a b
AI, I, I AT, T, T
eft traveling wave
ight traveling wave
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𝜕2𝑢
𝜕𝑡2= 𝐶0
2𝜕2𝑢
𝜕𝑥2 (6.5)
C0 is the elastic wave speed calculated as
𝐶0 = √𝐸
𝜌 (6.6)
D’Alembert solution applies to equation (6.6) and two quantities are of particular interest, the
stress σ ,t) and velocity V(x,t) at position x. Equation (6.7) give D’Alembert’s solution to
equation (6.5) for these quantities in the specimen in terms of stress waves travelling in the
positive and negative x direction.
𝑉(𝑥, 𝑡) =
𝐶𝑠
𝐸𝑠
(−𝜎𝑥(𝑥 − 𝐶0𝑡) + 𝜎−𝑥(𝑥 + 𝐶0𝑡))
𝜎(𝑥, 𝑡) = 𝐸𝑠(𝜎𝑥(𝑥 − 𝐶0𝑡) + 𝜎−𝑥(𝑥 + 𝐶0𝑡))
(6.7)
The total stress σ a,t) is calculated by summing all the incident and reflected waves at interface (a)
and equivalent for σ b,t at interface b . Three coefficients are defined to describe the
transmission and reflection of the wave:
B Transmission coefficient between the Incident bar and the Specimen
B1 Reflection coefficient between the specimen and the Incident bar
B2 Reflection coefficient between the specimen and the Transmitter bar
The specimen remains in perfect contact with the bars at all times so stress equilibrium over the
interface applies and no gaps are formed or superposition of material occurs. The coefficients are
calculated with classic wave mechanics as [3]
𝜎𝑠
𝜎𝐼
= 𝐵 =2𝐴𝐼𝜌𝑠𝐶𝑠
𝐴𝐼𝜌𝐼𝐶𝐼 + 𝐴𝑠𝜌𝑠𝐶𝑠
(6.8)
𝜎𝑠_𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑
𝜎𝑠
= 𝐵1 =𝐴𝐼𝜌𝐼𝐶𝐼 − 𝐴𝑠𝜌𝑠𝐶𝑠
𝐴𝑠𝜌𝑠𝐶𝑠 + 𝐴𝐼𝜌𝐼𝐶𝐼
(6.9)
𝜎𝑠_𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑
𝜎𝑠
= 𝐵2 =𝐴𝑇𝜌𝑇𝐶𝑇 − 𝐴𝑠𝜌𝑠𝐶𝑠
𝐴𝑠𝜌𝑠𝐶𝑠 + 𝐴𝑇𝜌𝑇𝐶𝑇
(6.10)
The specimen has length Ls and the time it for an elastic wave to propagate from the Incident
bar/specimen interface (a) to the specimen/Transmitter bar (b) is calculated by
𝑡0 =𝐿𝑠
𝐶𝑠
(6.11)
Each reflection at an interface is numbered by n , , ….k where k is the k’th reflection. The
Incident wave is defined as 𝜎𝐼(𝑡) with any arbitrary form. Figure 6.2 show a diagram over the
reflection process with time on the y-axis and position on the x-axis.
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Figure 6.2 Diagram for wave motion the specimen. 𝜎𝑛𝑅 is the reflected part of the wave at each reflection
between the wave and the interfaces.
The loading process starts at t=0 and n=0 where the incoming incident pulse 𝜎𝐼(𝑡)is reflected and
transmitted at the Incident bar/specimen interface (a). The transmitted part propagates through the
specimen heading towards interface (b). Until the reflected part from interface (b) comes back to
interface (a) the stress at interface (a) will be
n = 0 𝜎𝑎(𝑡) = 𝐵𝜎𝑖(𝑡), 0 ≤ 𝑡 < 2𝑡0 (6.12)
As the wave has to travel to and from interface (b), interface (a) will first see a change in its
loading condition after 2t0. As the wave reaches interface (b) at n=1 the wave reflects and
transmits and the stress is the sum of the incoming wave and the reflected part according to
equation (6.7). 𝜎𝑖(𝑡) is delayed t0 when it hits interface (b) the first time such that it becomes
1/Cs
Position
t
n=1
tBi
0
ttBBR 01 2
Ls
t0 =Ls/Cs
ttBBBR 2 0221
tB tBBR 32 0
2
31
tBB tBR 421 0
22
4
tBB tBR 521 0
32
5
tBB tBR 621 0
33
6
n=2
n=4
n=3
n=5
n=6
a b
n=0
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𝜎𝑖(𝑡 − 𝑡0). The incoming wave at interface (b) at t = t0 is then
𝜎𝑏 𝑖𝑛𝑐𝑜𝑚𝑖𝑛𝑔(𝑡) = 𝐵𝜎𝑖(𝑡 − 𝑡0) (6.13)
The wave is reflected and the reflected part (named 𝜎1𝑅 in Figure 6.2) is
𝜎1𝑅(𝑡) = 𝐵 ∙ 𝐵2 ∙ 𝜎𝑖(𝑡 − 𝑡0) (6.14)
The sum of the incoming and reflected wave gives the total stress at interface (b) as
n = 1
𝜎𝑏(𝑡) = 𝐵𝜎𝑖(𝑡 − 𝑡0) + 𝐵 ∙ 𝐵2 ∙ 𝜎𝑖(𝑡 − 𝑡0) ⟺ 𝜎𝑏(𝑡) = 𝐵 ∙ (1 + 𝐵2) ∙ 𝜎𝑖(𝑡 − 𝑡0) 𝑡0 ≤ 𝑡 < 3𝑡0
(6.15)
This sum is maintained as a function of 𝜎𝑖(𝑡) until the wave reaches back from interface (a) again.
𝜎1𝑅 propagates back towards interface (a) where it arrives at time t = 2t0. The wave reflects and
transmits and with the reflection coefficient B1 of interface (a) the reflected part becomes
𝜎2𝑅(𝑡) = 𝐵 ∙ 𝐵1 ∙ 𝐵2 ∙ 𝜎𝑖(𝑡 − 2𝑡0) (6.16)
The total stress at interface (a) is now the sum of the incoming wave 𝜎1𝑅 from interface (b) plus
the reflection 𝜎2𝑅 and further also the transmitted part of the incident wave from the Incident bar
which is still is present at the interface at time t. 𝜎2𝑅 is also the Incident wave, just transmitted,
reflected twice and delayed 2t0. The total stress is calculated as
n = 2
𝜎𝑎(𝑡) = 𝐵𝜎𝑖(𝑡) + 𝐵 ∙ 𝐵2 ∙ 𝜎𝑖(𝑡 − 2𝑡0) + 𝐵 ∙ 𝐵1 ∙ 𝐵2 ∙ 𝜎𝑖(𝑡 − 2𝑡0) ⟺ 𝜎𝑎(𝑡) = 𝐵 ∙ 𝜎𝑖(𝑡) + 𝐵 ∙ (𝐵2 + 𝐵1𝐵2) ∙ 𝜎𝑖(𝑡 − 2𝑡0) 2𝑡0 ≤ 𝑡 < 4𝑡0
(6.17)
𝜎2𝑅 propagates towards interface (b) where it is the incoming wave that reflects and transmits. The
reflected part becomes
𝜎3𝑅(𝑡) = 𝐵 ∙ 𝐵1 ∙ 𝐵22 ∙ 𝜎𝑖(𝑡 − 3𝑡0) (6.18)
The total stress at interface (b) is the sum of all incoming and reflected parts with their respective
time delays
n = 3
𝜎𝑏(𝑡) = 𝐵𝜎𝑖(𝑡 − 𝑡0) + 𝐵 ∙ 𝐵2 ∙ 𝜎𝑖(𝑡 − 𝑡0) + 𝐵 ∙ 𝐵1 ∙ 𝐵2 ∙ 𝜎𝑖(𝑡 − 3𝑡0) + 𝐵
∙ 𝐵1 ∙ 𝐵22 ∙ 𝜎𝑖(𝑡 − 3𝑡0) ⟺
𝜎𝑏(𝑡) = 𝐵 ∙ (1 + 𝐵2) ∙ 𝜎(𝑡 − 𝑡0) + 𝐵 ∙ (𝐵1𝐵2 + 𝐵1𝐵2
2) ∙ 𝜎(𝑡 − 3𝑡0) 3𝑡0 ≤ 𝑡 < 5𝑡0
(6.19)
𝜎3𝑅 becomes the incoming wave at interface (a) and the reflected part of 𝜎3𝑅 at interface (a) is
𝜎4𝑅(𝑡) = 𝐵 ∙ 𝐵12 ∙ 𝐵2
2 ∙ 𝜎𝑖(𝑡 − 4𝑡0) (6.20)
The total sum of stress contributions at interface (a) now becomes
n = 4
𝜎𝑎(𝑡) = 𝐵𝜎𝑖(𝑡) + 𝐵 ∙ 𝐵2 ∙ 𝜎𝑖(𝑡 − 2𝑡0) + 𝐵 ∙ 𝐵1 ∙ 𝐵2 ∙ 𝜎𝑖(𝑡 − 2𝑡0) + 𝐵 ∙ 𝐵1
∙ 𝐵22 ∙ 𝜎𝑖(𝑡 − 4𝑡0) + 𝐵 ∙ 𝐵1
2 ∙ 𝐵22 ∙ 𝜎𝑖(𝑡 − 4𝑡0)
𝜎𝑎(𝑡) = 𝐵𝜎𝑖(𝑡) + 𝐵 ∙ (𝐵2 + 𝐵1𝐵2) ∙ 𝜎𝑖(𝑡 − 2𝑡0) + 𝐵 ∙ (𝐵1𝐵2
2 + 𝐵12𝐵2
2)∙ 𝜎𝑖(𝑡 − 4𝑡0)
4𝑡0 ≤ 𝑡 < 6𝑡0
(6.21)
Again the reflected part, now 𝜎4𝑅, of the incoming wave, propagates towards interface (b). At n=5
the added incoming and reflected part is
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n = 5 𝜎𝑏(𝑡) =. . . +𝐵 ∙ (𝐵2
2𝐵22 + 𝐵2
2𝐵23) ∙ 𝜎𝑖(𝑡 − 5𝑡0)
5𝑡0 ≤ 𝑡 < 7𝑡0 (6.22)
This process continues as long the Incident wave is present and, at a certain time, the wave is
reflected so many times that there is no practical addition to the stress state in the interfaces. For
calculations beyond 5 transits, equations (6.12) to (6.22) is rewritten into a sum function of the
wave for interface (a)
𝜎𝑠(𝑎, 𝑡) = 𝐵 ∑ 𝐹(𝑛, 𝐵1, 𝐵2)𝜎𝐼(𝑡 − 𝑛𝑡0)
⌊𝑡
𝑡0⌋
𝑛=0
(6.23)
⌊𝑡
𝑡0⌋ is the largest possible integer (n) of the ratio
𝑡
𝑡0 and F(n) is a coefficient vector. From equations
(6.12) and (6.22), F(n) is written as
𝐹(𝑛, 𝐵1, 𝐵2) = [𝐵1𝑥1(𝑛)
𝐵2𝑥2(𝑛)
+ 𝐵1𝑥2(𝑛)
𝐵2𝑥3(𝑛)
], 𝑛 > 0 (6.24)
Equations (6.12) - (6.22) show the exponents x1, x2 and x3 develops as function of n as given in
Table 6.2.
Table 6.2 Sequences for exponents in equation (6.24) for n=0 to 10
n x1 x2 x3
0 0 0 0
1 0 0 1
2 0 1 1
3 1 1 2
4 1 2 2
5 2 2 3
6 2 3 3
7 3 3 4
8 3 4 4
9 4 4 5
10 4 5 5
x1 to x3 are expressed as a function of n as
𝑥1(𝑛) =𝑛 − 1
2+
1𝑛 + (−1)𝑛
4
𝑥2(𝑛) =𝑛
2+
1𝑛 + (−1)𝑛
4
𝑥3(𝑛) =𝑛 + 1
2+
1𝑛 + (−1)𝑛
4
(6.25)
Equation (6.24) is only valid for n>0. In the case where n=0, F(n) equals 2, but F(n) should be 1
according to Equation (6.12) as only the transmitted wave is present at t = 0-2t0. So explicitly F(0)
= 0.
Equations (6.12) and (6.22) also show interface (a) only gets an extra added term for even numbers
of n while interface (b) gets an extra added term for odd numbers of n. Then the interface stresses
are written as
𝜎𝑠(𝑎, 𝑡) = 𝐵 ∑ 𝐹(𝑛, 𝐵1, 𝐵2)𝜎𝐼(𝑡 − 𝑛𝑡0)
⌊𝑡
𝑡0⌋
𝑛=0
, 𝑛 = 0,2,4 … . 𝑘 (6.26)
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𝜎𝑠(𝑏, 𝑡) = 𝐵 ∑ 𝐹(𝑛, 𝐵1, 𝐵2)𝜎𝐼(𝑡 − 𝑛𝑡0)
⌊𝑡
𝑡0⌋
𝑛=0
, 𝑛 = 1,3,5 … . 𝑘 (6.27)
The structure of F(n) is now exploited to derive equations for the velocity at the interfaces. First, it
is necessary to recap from equation (6.7) that for the velocity, waves traveling in the positive x
direction had negative sign in the sum while waves traveling in the negative x direction had a
positive sign in the sum. For the stress computation, both waves had a positive sign. Examination
of F(n) shows it is calculated as the product of B1 and B2 with one set of exponents, and this set is
added with the other product of B1 and B2, which has another set of exponents. Careful
examination of equations (6.12) to (6.22) reveals that the first product describes the incoming
waves to the interfaces at their respective wave transit number (n). The second product describes
the reflected wave at the interfaces and the reflected waves are always altered by an extra
reflection compared to their incoming counterpart. This explains why the second product in F(n) is
“ahead” in the e ponents compared to the first product of F n . Figure 6.3 shows that interface (a)
has its incoming waves traveling in the negative x direction while the reflected waves travel in the
positive x direction. For interface (b) it is the opposite and thus the interfaces require a different set
of signs to calculate the velocity according to equation (6.3).
Figure 6.3 Wave directions.
Thus two coefficient functions Fva(n) and Fvb(n) are defined and with negative signs for the
reflected waves, Fva(n) becomes
𝐹𝑣𝑎(𝑛, 𝐵1, 𝐵2) = [𝐵1𝑥1(𝑛)
𝐵2𝑥2(𝑛)
− 𝐵1𝑥2(𝑛)
𝐵2𝑥3(𝑛)
], 𝑛 = 2,4,6 … 𝑘 (6.28)
With negative sign for incoming waves, Fvb(n) becomes
𝐹𝑣𝑏(𝑛, 𝐵1, 𝐵2) = [−𝐵1𝑥1(𝑛)
𝐵2𝑥2(𝑛)
+ 𝐵1𝑥2(𝑛)
𝐵2𝑥3(𝑛)
], 𝑛 = 1,3,5 … 𝑘 (6.29)
As with F(n), Fva(n) and Fvb(n) are only defined for n>0. Fva(0) is -1 as the transmitted wave
passes through the interface in the positive x direction. Fvb(0) is not used, but is set to 1. The
velocity is now calculated in a similar way as the stresses in equation (6.26) and (6.27) using
D’Alemberts solution.
𝑉𝑠(𝑎, 𝑡) =
𝐶𝑠
𝐸𝑠
𝐵 ∑ 𝐹𝑣𝑎(𝑛, 𝐵1 , 𝐵2)𝜎𝐼(𝑡 − 𝑛𝑡0)
⌊𝑡
𝑡0⌋
𝑛=0
, 𝑛 = 0,2,4 … . 𝑘 (6.30)
a
a
b
b
a b
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𝑉𝑠(𝑏, 𝑡) =
𝐶𝑠
𝐸𝑠
𝐵 ∑ 𝐹𝑣𝑏(𝑛, 𝐵1 , 𝐵2)𝜎𝐼(𝑡 − 𝑛𝑡0)
⌊𝑡
𝑡0⌋
𝑛=0
, 𝑛 = 1,3,5 … . 𝑘 (6.31)
Equations (6.26), (6.27), (6.30) and (6.31) form a set of equations that describe the variation of
stress and velocity at the specimen ends for any arbitrary chosen incident wave and impedance
mismatch between the bars and specimen. Thus, the equilibrium process can be estimated for any
combination of bars, specimen and Incident wave.
When the specimen stress is known, the reflected wave in the Incident bar can be calculated, as
well as the transmitted wave in the Transmitter bar. Applying the assumption of force equilibrium
at interface (a) yields
𝐴𝐼(𝜎𝐼(𝑡) + 𝜎𝑅(𝑡)) = 𝐴𝑠𝜎𝑠(𝑎, 𝑡) ⇔
𝜎𝑅(𝑡) =𝐴𝑠
𝐴𝐼
𝜎𝑠(𝑎, 𝑡) − 𝜎𝐼(𝑡) (6.32)
With force equilibrium over interface (b) the transmitted wave is calculated as
𝐴𝑆𝜎𝑠(𝑏, 𝑡) = 𝐴𝑡𝜎𝑡(𝑡)
𝜎𝑇(𝑡) =𝐴𝑆
𝐴𝑇
𝜎𝑠(𝑏, 𝑡) (6.33)
The Incident, Reflected and Transmitted waves are known for the Incident and Transmitter bar,
and the SHPB formulas can be applied to calculate the linear stress strain curve. More importantly,
the required stress wave amplitude in the bars can be estimated and compared to the yield strength
of the bars as well as measurement system capacity.
6.3 Validation of the model
The GW model is compared to the wave mechanic model presented by Frew et al. [20, 21]
(Equation (9) and (10)). Adapted from Yang et al. [19] three Incident waves is used to compare the
models: a step pulse, a step pulse with finite rise time, and a linear rising pulse. The waves are
shown in Figure 6.4A. The model derived by Frew et al. is only valid for equal bars so B1 = B2 is
used for the GW model. Both models are implemented in MatLab to compare the stresses at
interface (a) and (b). The average stress difference between the models is calculated as
∆𝜎̅̅̅̅ =(𝜎(𝑎, 𝑡) − 𝜎𝐹𝑟𝑒𝑤(𝑎, 𝑡)) + (𝜎(𝑏, 𝑡) − 𝜎𝐹𝑟𝑒𝑤(𝑏, 𝑡))
2 (6.34)
Figure 6.4B shows equation (6.34) plotted as a function of wave transit in the specimen for the
three loading pulses shown in Figure 6.4A. The average stress difference is measured in µPa,
which in practice mean the models behave equally.
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A) Waves pulses used for comparison of the models B) Interface stress σa and σb
Figure 6.4 Comparison of the wave mechanics model (Equations 9 and 10 in [21]) presented by Frew et al.
and the GW model derived in this work (Equations (6.26) and (6.27))
Yang et al. [19] presented formulas to calculate the equilibrium state as a function of the number
of wave transits n for different Incident waves. The formulas are only valid for equal bars so B1 =
B2 in this case also. The equilibrium state is calculated as in equation (6.1). Figure 6.5 shows a
comparison of the GW model and the formulas derived by Yang et al.
Step pulse 𝑅(𝑛) =2𝛽(1 − 𝛽)𝑛−1
(1 + 𝛽)𝑛 − (1 − 𝛽)𝑛
Finite rise time
pulse 𝑅(𝑛) =
2𝛽(1 − 𝛽)𝑛−2
(1 + 𝛽)𝑛 − (1 − 𝛽)𝑛−2, 𝑘 > 2
Linear rising
pulse 𝑅(𝑛) =
2𝛽2 [1 − (−(1 − 𝛽)
1 + 𝛽)
𝑛
]
2𝑛𝛽 − 1 + (1 − 𝛽1 + 𝛽
)𝑛 , 𝑘 > 2
𝛽 =𝐴𝐼𝑍𝐼
𝐴𝑠𝑍𝑠
A) Formulas for calculating equilibrium measure as
function of number of wave transits (n). B) R(n) for β=0.0122 compared with the GW model
Figure 6.5 Comparison between the GW model and the formulas derived by Yang et al. [19].
For the linear rising pulses, the models yields the same result while for step pulse the model from
Yang et al. predict a value between two time steps in the GW model. For the step pulse with a
finite rise time, the model by Yang et al. predict a faster equilibrium process compared to the GW
model. The model by Yang et al. is only defined for n > 2 and the finite rise time in the pulse is set
to 2n for their derivation. A pulse with finite rise of 2n has been used for both the models so the
-5 0 5 10 15 20 250
100
200
300
400
500
n = t/t0
Pu
lse
amp
litu
de
(MPa
)
Step pulse
Finite rise time pulse
Linear rising pulse
0 5 10 15 200
0.5
1
1.5
2
n = t/t0
Ave
rage
str
ess
dif
fere
nce
(
Pa)
Step pulse
Finite rise time pulse
Linear rising pulse
0 5 10 15 20 25 300
2
4
6
8
10
n = t/t0
RE
Q (%
)
Yang 2005 - Step pulse
Yang 2005 - Finite rise time pulse
Yang 2005 - Linear rising pulse
Wave model - Step pulse
Wave model - Finite rise time pulse
Wave model - Linear rising pulse
Department of Mechanical Engineering
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difference is in the formulation of the models.
6.4 Effect of the Incident wave
The GW model can be used with any arbitrary loading pulse to predict the evolution of stress
equilibrium and strain rate with any set of combination of bars and specimen. To examine the
effect of the general shape of the Incident wave on the evolution in stress equilibrium and strain
rate, the three loading pulses presented in Figure 6.4A are used along with an exponential growing
curve 𝐹(𝑥) = 𝐶𝑡𝑥 with x≠1. The loading pulses are shown in Figure 6.6 and they represent the set
of loading pulse shapes, which can be generated in a SHPB test rig.
Figure 6.6 Different loading pulses
The influence of the shape of the loading pulse on the evolution of stress equilibrium and strain
rate is examined with B1 = B2 = 0.667 in Figure 6.7. Dynamic equilibrium and constant strain rate
are only reached with the linear rising pulse. In Figure 6.8 the case with rigid boundaries (B1 = B2
= 1) is examined. Here the step pulse with a finite rise time gives the best result, and the
equilibrium evolution follows that of the linear rising pulse until the pulse reaches constant
amplitude. The finite rise ensures faster stress equilibrium and reaches the same strain rate as the
pure step pulse. None of the other loading pulses are able to create a state of constant strain rate
with rigid boundaries. Thus if the test specimen is situated between elastic boundaries as in the
SHPB test rig, and a state of stress equilibrium and constant strain are sought, then the linear rising
pulse is the best option. Figure 6.9 shows the same holds true if the bars are made with different
impedances. If the boundaries are close to rigid, a step pulse with a finite rise time is the best
option.
-5 0 5 10 15 20 250
100
200
300
400
500
n = t/t0
Puls
e am
plit
ud
e (M
Pa)
Step pulse
Finite rise time pulse
Linear rising pulse
Exponential pulse, x>1
Exponential pulse x<1
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A) Stress equilibrium B) Strain rate
Figure 6.7 Simulation of stress equilibrium and strain rate evolution for B1 = B2 = 0.667
A) Stress equilibrium B) Strain rate
Figure 6.8 Simulation of stress equilibrium and strain rate evolution for B1 = B2 = 1. (Rigid boundaries)
A) Stress equilibrium B) Strain rate
Figure 6.9 Simulation of stress equilibrium and strain rate evolution for B1 = 0.667, B2 = 0.22.
0 5 10 150
5
10
15
20
25
n = t/t0
RE
Q (%
)
Step pulse
Finite rise time pulse
Linear rising pulse
Exponential pulse, x>1
Exponential pulse x<1
0 5 10 150
500
1000
1500
n = t/t0
Stra
in r
ate
(/s)
Step pulse
Finite rise time pulse
Linear rising pulse
Exponential pulse, x>1
Exponential pulse x<1
0 5 10 150
5
10
15
20
25
n = t/t0
RE
Q (%
)
Step pulse
Finite rise time pulse
Linear rising pulse
Exponential pulse, x>1
Exponential pulse x<1
0 5 10 150
500
1000
1500
n = t/t0
Stra
in r
ate
(/s)
Step pulse
Finite rise time pulse
Linear rising pulse
Exponential pulse, x>1
Exponential pulse x<1
0 5 10 150
5
10
15
20
25
n = t/t0
RE
Q (%
)
Step pulse
Finite rise time pulse
Linear rising pulse
Exponential pulse, x>1
Exponential pulse x<1
0 5 10 150
500
1000
1500
n = t/t0
Stra
in r
ate
(/s)
Step pulse
Finite rise time pulse
Linear rising pulse
Exponential pulse, x>1
Exponential pulse x<1
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The linear rising pulse as the best option for linear elastic brittle specimens was mentioned in
several papers [17, 18, 20, 69, 71, 74, 83, 96, 97], but only for ensuring faster stress equilibrium.
6.5 Stress equilibrium and constant strain rate
The reflection function F(n) is exploited to give information about what specimen size, type of
bars, and constant loading rate should be used to ensure that both stress equilibrium, and constant
strain rate occur before the specimen fails. Equations (6.26) and (6.27) give the stress at the
respective interfaces of the specimen for an arbitrary stress pulse σI as a function of time.
However, based on the finding in section 6.4 the loading pulse σI is restricted to a linear rising
stress pulse, with slope M, given as 𝜎𝐼(𝑡) = 𝑀 ∙ 𝑡. If the stress is evaluated only at the k’th
reflection such that 𝑡 = 𝑘𝑡0 then equation (6.26) is written as
𝜎(𝑎, 𝑘) = 𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑ 𝐹(𝑛, 𝐵1 , 𝐵2)(𝑘 − 𝑛)
𝑘
𝑛=0
, 𝑛 = 0,2,4 … 𝑘 (6.35)
Equation (6.27) becomes
𝜎(𝑏, 𝑘) = 𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑ 𝐹(𝑛, 𝐵1, 𝐵2)(𝑘 − 𝑛)
𝑘
𝑛=0
, 𝑛 = 1,3,5 … 𝑘 (6.36)
In the SHPB community it is widely accepted that the specimen is in dynamic equilibrium when
REQ (equation (6.1)) is less than 5% [17-19, 22, 69]. So inserting equations (6.35) and (6.36) into
(6.1) yields
𝑅𝑒𝑞(𝑘)
= 2 ∙ |𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑ 𝐹(𝑛, 𝐵1, 𝐵2)(𝑘 − 𝑛)𝑘
𝑛=0 − 𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑ 𝐹(𝑛, 𝐵1, 𝐵2)(𝑘 − 𝑛)𝑘𝑛=0
𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑ 𝐹(𝑛, 𝐵1, 𝐵2)(𝑘 − 𝑛)𝑘𝑛=0 + 𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑ 𝐹(𝑛, 𝐵1, 𝐵2)(𝑘 − 𝑛)𝑘
𝑛=0
| ⇔
𝑅𝑒𝑞(𝑘) = 2 ∙ |∑ 𝐹(𝑛, 𝐵1, 𝐵2)(𝑘 − 𝑛)𝑘
𝑛=0,2… − ∑ 𝐹(𝑛, 𝐵1 , 𝐵2)(𝑘 − 𝑛)𝑘𝑛=1,3…
∑ 𝐹(𝑛, 𝐵1, 𝐵2)(𝑘 − 𝑛)𝑘𝑛=0,2… + ∑ 𝐹(𝑛, 𝐵1 , 𝐵2)(𝑘 − 𝑛)𝑘
𝑛=1,3...
|
(6.37)
Thus, with the assumption of a linear rising loading pulse, the stress equilibrium is independent of
the stress rate, the specimen length, and the wave velocity. This is consistent with the findings for
the case of equal bars [17, 94].
The strain rate in the specimen is linked to Fva(n) and Fvb(n). Equations (6.31) and (6.32) are
rewritten in the same way as equation (6.35) and (6.36) to obtain the velocity in the specimen at
the interfaces
𝑉𝑠(𝑎, 𝑘) =𝐶𝑠
𝐸𝑠
∙ 𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑ 𝐹𝑣𝑎(𝑛)(𝑘 − 𝑛)
𝑘
𝑛=0
, 𝑛 = 0,1,2 … 𝑘 (6.38)
𝑉𝑠(𝑏, 𝑘) =𝐶𝑠
𝐸𝑠
∙ 𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑ 𝐹𝑣𝑏(𝑛)(𝑘 − 𝑛)
𝑘
𝑛=0
, 𝑛 = 0,1,2 … 𝑘 (6.39)
The average strain rate over a specimen of length Ls settled between two bars is calculated as [3]:
𝑑𝜀
𝑑𝑡= 𝜀̇(𝑡) =
𝑉𝑠(𝑎, 𝑡) − 𝑉𝑠(𝑏, 𝑡)
𝐿𝑠
(6.40)
Inserting equations (6.38) and (6.39) into (6.40) yield
𝜀̇(𝑡) =1
𝐿𝑠
∙𝐶𝑠
𝐸𝑠
∙ 𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑(𝑘 − 𝑛) ∙ (𝐹𝑣𝑎(𝑛, 𝐵1, 𝐵2) − 𝐹𝑣𝑏(𝑛, 𝐵1, 𝐵2))
𝑘
𝑛=0
(6.41)
Utilizing that 𝑡0 = 𝐿𝑠/𝐶𝑠 the strain rate is calculated as
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𝜀̇(𝑘) =1
𝐸𝑠
∙ 𝐵 ∙ 𝑀 ∙ ∑(𝑘 − 𝑛) ∙ 𝐹𝑣(𝑛, 𝐵1, 𝐵2)
𝑘
𝑛=0
(6.42)
Where Fv(n) is calculated as the sum of Fva and Fvb remembering Fva is only defined for even
numbers and Fvb is only defined for odd numbers. Then Fv becomes
𝐹𝑣(𝑛, 𝐵1, 𝐵2) = [𝐵1𝑥1(𝑛)
𝐵2𝑥2(𝑛)
− 𝐵1𝑥2(𝑛)
𝐵2𝑥3(𝑛)
], 𝑛 = 1,2,3 … 𝑘 (6.43)
Equation (6.43 shows that the strain rate is proportional to the convolution sum of Fv and (k-n).
The strain rate will reach a constant level when the convolution sum do not increase anymore with
increasing values of k, which will happen when Fv(n) is small. The bars must have greater
mechanical impedance than the specimen to ensure loading of the specimen, so B1 and B2 must be
greater than 0. The maximum value of B1 and B2 is one2, which is completely rigid bars. An
analyses of Fv for n = 500 and all combinations of B1 and B2 between 0 and 1 gives a max value
of 8x10-4
at B1 = B2 =1. Then, at n=500 all combinations of B1 and B2 would have reached
constant strain rate, and n = 500 can be used as an estimate of the maximum strain rate. The
relative deviation from constant strain rate is expressed as
𝑅𝑆𝑅(𝑘, 𝐵1, 𝐵2) =∑ (500 − 𝑛) ∙ 𝐹𝑣(𝑛, 𝐵1, 𝐵2)
500
𝑛=0− ∑ (𝑘 − 𝑛) ∙ 𝐹𝑣(𝑛, 𝐵1, 𝐵2)
𝑘
𝑛=0
∑ (500 − 𝑛) ∙ 𝐹𝑣(𝑛, 𝐵1, 𝐵2)500
𝑛=0
(6.44)
When RSR is below 5%, a dynamic constant strain can be said to have occurred. Note that constant
strain rate only occurs when Va(t) ≠ Vb(t) and Va(t) and Vb(t) are constant. Alternatively, if Va(t) ≠
Vb(t) and Va(t) and Vb(t) are undergoing the same acceleration, the velocity difference is constant,
even though the velocity changes on both sides of the specimen. The number of reflections to
constant strain rate is independent of t0 and the maximum strain rate is then not a function of the
specimen length. However, the specimen length controls how long it takes to reach the maximum
strain rate. The well-known estimator of strain rate given in equation (6.45) is thus not useable
with linear elastic materials loaded by linear rising pulse.
𝜀�̇�𝑛𝑔 =𝑉
𝐿0
(6.45)
It is interesting to examine whichever stress equilibrium or constant strain rate occurs first.
Equations (6.37) and (6.44) are evaluated for all combinations of B1 and B2 to find n which
satisfy both REQ and RSR < 0.05. The result is plotted in Figure 6.10A as a contour plot. The blue
dot represents a setup with B1 = B2 = 0.5. For this combination, stress equilibrium will happen
between 8 and 10 transits of the wave through the specimen while constant strain rate is achieved
after 4 to 6 transits. The diagonal of Figure 6.10A corresponds to equals bars and is plotted in
Figure 6.10B. Whichever constant strain rate or stress equilibrium is reached first depends on the
combination of impedance mismatches, also for equal bars. This is inconsistent with Foster et al.
[94] who found that stress equilibrium would always happen first. However, this result was
obtained from a homogenous approach and not the full wave mechanics approach as here.
2 See Appendix B.
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A) Contour plot of wave transits to obtain dynamic
equilibrium and constant strain rate
B) Number of reflections to constant strain rate with
equal bars
Figure 6.10 Number of reflections in the specimen to achieve stress equilibrium and constant strain rate
Figure 6.10 does not indicate if stress equilibrium or constant strain rate is fulfilled before
specimen failure for a particular test. In addition, it does not indicate the achievable strain rate
before failure. Figure 6.10 only indicates how many round trips the wave requires to equilibrate
the condition. The final strain rate is determined from the applied stress rate, B1 and B2.
6.6 Test design
The GW model is well suited for designing a test in the Split Hopkinson Pressure Bar for FRP
materials, if they are assumed linear elastic. In most cases, an SHPB is already available and
maybe with Incident and Transmitter bars made from different materials. The challenge is to select
the correct specimen size, together with the correct stress rate of the incident pulse to ensure
conditions of stress equilibrium and constant strain rate before failure. If the elastic modulus and
density are known for both the bars and the test specimen, the design process narrows to estimate
the specimen cross sectional area As along with the length Ls. First for a given set of A and Ls, the
number of wave transit to stress equilibrium (NEQ) and constant strain rate (NSR) should be
calculated. Then the greater of these (nmax=max{NEQ,NSR}) is used to solve equation (6.46) for the
stress rate M.
𝜎𝑠_𝑓𝑎𝑖𝑙𝑢𝑟𝑒 = 𝜎(𝑎, 𝑛𝑚𝑎𝑥) = 𝐵 ∙ 𝑀 ∙ 𝑡0 ∙ ∑ 𝐹(𝑛)(𝑛𝑚𝑎𝑥 − 𝑛)
𝑛𝑚𝑎𝑥
𝑛=0
(6.46)
𝜎(𝑎, 𝑘) is used since the stress at interface (a) will always be larger than at interface (b). When M
is calculated, the matching maximum strain rate is calculated from equation (6.42). Thus, the
maximum strain rate, the maximum stress rate, and the time to fracture are known. Further
analyses could include the required pulse length and maximum amplitude in the Incident bar.
Finally, the required pulse can then be matched through the available pulse shaping technique. The
design algorithm is summarised in table 6.3.
0 0.2 0.4 0.6 0.8 1 0
0.2
0.4
0.6
0.8
1
2 2
4 6
8
B2
B1
Constant strain rate Stress equilibrium
0 0.2 0.4 0.6 0.8 1 0
5
10
15
20
B1 = B2
n=t
/to
Constant strain rate Stress equilibrium
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Table 6.3 SHPB specimen design algorithm
Step Description
0 Select bars and set the design space for Ls and A. For example Ls can vary between 3 and 15mm. Also select criteria for constant strain rate and stress equilibrium
Steps 1 to 8 are repeated for each combination of A and Ls
1 Calculate transmission and reflection coefficient B, B1 and B2
2 Calculate F and Fv
3 Create Fa
4 Find number (NSR) of wave transits to constant strain rate
5 Find number (NEQ) of wave transits to stress equilibrium
6 Select the highest number of NSR and NEQ
7 Calculate maximum stress rate
8 Calculate achievable constant strain rate using the result from step 7
9 Check of maximum stress in the Incident bar and Transmitter bar.
Also check the required wave duration
As example the design algorithm with is used for a composite with an elastic modulus of 8.7 GPa
in the loading direction and a density 2000kg/m3. The SHPB is equipped with a 25mm diameter
Incident bar made from steel with E = 191GPa and ρ = 7741kg/m3. The Transmitter bar is 25mm
in diameter, E = 73.4GPa and ρ = 2809kg/m3. The specimen design is restricted such that the cross
sectional area can vary in fractions between 0.2 and 0.8 of the bar cross sectional area. The gage
length is constrained to lie in the interval 3-15mm.
Figure 6.11 gives maximum strain rate and stress rate as contour plots, calculated using the design
algorithm in
Department of Mechanical Engineering
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Table 6.3. For example an area ratio of 0.25 and length of 10 mm will give an approximate
maximum strain rate of 200 /s.
A) Maximum strain rate as a function of specimen
cross section ratio As/AB and specimen length Ls.
B) Maximum stress rate in the Incident bar as a
function of specimen cross section ratio As/AB and
specimen length Ls.
Figure 6.11 Maximum strain rate and stress rate
The blue dot in Figure 6.11A represents a specimen with a square cross section with dimensions
12x12mm and a length of 10mm. This specimen was tested in the described SHPB and the strain –
strain rate curves are shown in Figure 6.12A for different stress rates in the Incident wave. The
corresponding incident waves are shown in Figure 6.12B. The test is described in details in chapter
8. A plateau is realised followed by a steady rise at low strain rate. The plateau is in agreement
with Figure 6.11A, which predict that a steady strain rate of 150/s is possible. The limiting stress
rate at the blue dot is 1.17 MPa/µs and the stress rate was 0.91 MPa/µs at low strain rate as shown
in Table 6.4. The subsequent rise in strain rate from the plateau is believed to be due to the
nonlinearity in the Incident wave, and failure progression in the specimen. The medium strain rate
is around 250-300/s with a stress rate of around 1.42 MPa/µs. This is higher than the calculated
limit and is indicated by slightly varying strain rate throughout the entire deformation history. For
the high strain rate the stress rate is nearly 3 times higher than the calculated limit and this is seen
as a highly varying strain rate throughout the deformation history.
4 6 8 10 12 140.2
0.3
0.4
0.5
0.6
0.7
0.8
200
200
250
250
250
300
30030
0
300
30
0
400
400
400
40
0
400
600
60
0
600
600
80
0
80
0
80
0
10
00
10
00
Ls (mm)
As/A
B
Max strain rate (/s)
4 6 8 10 12 140.2
0.3
0.4
0.5
0.6
0.7
0.8
1
1
1
1.4
1.4
1.4
2
2
2
2
3
3
3
3
5
5
5
5
8
8
8
10
10
Ls (mm)
As/A
B
Max stress rate (MPa/s)
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A) Strain rate vs. strain curves B) Incident waves with different stress rates in the
rising part
Figure 6.12 Incident waves and strain rate vs. strain curves for the design case
Table 6.4 Stress rates in Incident pulses in Figure 6.12
Strain rate level Stress rate (MPa/µs) Measurement interval (µs)
Low 0.91 100-150
Medium 1.42 50-100
High 3.16 50-75
6.7 Radial inertia - considerations
Radial inertia should also be considered when test specimens are deformed at high strain rates. The
radial inertia due to the Poisson ratio will cause an added stress increment in the specimen, which
causes an error in the stress measurement. Forrestal et al. [98] derived a closed form solution for
the added inertia stress for an isotropic linear elastic material deforming at stress equilibrium. The
addition of the stress in the loading direction is given by equation (6.47) [98] where “ν” is
Poisson’s ratio, “a” is the radius of the specimen, “ρ” the specimen density, and “r” is the position
along the radial direction of the specimen. The added stress in the loading direction is proportional
to the change of strain rate and the same is the case in the radial direction.
𝜎𝑥 =𝜈𝑠
2(3 − 2𝜐𝑠)
4(1 − 𝜐𝑠)[𝑎𝑠
2 −2𝑟2
(3 − 2𝜐𝑠)] 𝜌𝑠
𝑑2𝜀𝑥0
𝑑𝑡2 (6.47)
Thus if the specimen is deforming a constant strain rate the inertial stress increase will vanish.
Further Forrestal and Chen et al. [18, 98] concluded that the for the strain rates which are
generated in a SHPB rig the magnitude of the added stress is 1MPa or below. Thus, radial inertia
can be ignored for high performance fibre reinforced specimens, and the effect is not included in
the GW model.
6.8 Simulation tool
The formulation of equation (6.27) and (6.26) allows any arbitrary loading pulse to be analysed
with the GW model. For example a simulated pulse-shaped loading pulse could be fed into the
model, or an actual recorded loading pulse from a SHPB. No loading pulse is perfectly linear, so it
is of interest to see the influence of the varying loading pulse on the simulated specimen response.
0 0.5 1 1.5 2 2.5 3 3.50
100
200
300
400
500
600Strain rates
Strain (%)
Stra
inra
te (/
s)
Low strain rate
Medium strain rate
High strain rate
0 50 100 150 200 250 3000
50
100
150
200
250
Time (s)
Inci
den
t b
ar s
tres
s (M
Pa)
Low strain rate
Medium strain rate
High strain rate
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Figure 6.13 gives an overview of the simulation process with the GW model.
Figure 6.13 Schematic of the simulation model
The Incident, Reflected and Transmitted waves in the bars are calculated from equations (6.32)
and (6.33) and can be used to post process the simulation with the different post processing
equations as well as verifying stress equilibrium and constant strain rate. Figure 6.14 and Figure
6.15 show simulation and comparison for the two cases in Figure 6.12, “ ow” and “high” strain
rate. The pulse shaper code presented in chapter 5 was used to simulate the Incident waves. The
stress strain curves were created by first calculating σ(a,t), σ(b,t), V(a,t) and V(b,t) and then
calculating the average stress, and strain rate and strain from equation (6.40).
First, consider the low strain rate test in Figure 6.14. The test was done with the setup described in
section 6.6. In the stress strain curves from the real test in Figure 6.14B, there is an offset in strain
before the stress starts to rise. This is due to a small air gap between the bars and the specimen,
which allow the bars to move without loading the specimen. Normally the specimen is kept in
place with some grease and the strain offset of 0.19% correspond to 20µm, which is likely to be
grease and air. The pre-simulate stress strain curve is created by running the simulated Incident
wave through the GW model and calculating the stress, strain rate and strain. The “Post-simulated”
curve in the strain rate plot was created by running the actual measured Incident wave through the
GW model. The Pre-simulated model was offset with 0.19% strain to match the offset of the stress
strain curves from the real tests. There is agreement between the simulation and the actual
measured stress strain response. The simulated Incident wave has the same shape as the measured
wave; however, there is a deviation in amplitude. The beginning of the measured wave lacks signs
of yielding of the pulse shaper, which indicating some damping in the beginning of the initial
elastic response of the pulse shaper. The measured stress equilibrium is above the simulated curve,
but the measured stress equilibrium response does come below 5% relative deviation. The
measured strain rate is actually the most stable compared to the simulated strain rates. The pre-
simulation predicts first a stable region, and then an increasing strain rate. The increase is due to
the Incident wave, which never becomes linear with the use of plastically deforming pulse-
shapers. Therefore, at least a part of the increase in the strain rate curve can be explained from the
input pulse, which changes loading rate as seen in Figure 6.14A.
Generate loading pulse
Simulate wave motion in
specimen
erify condition of setup
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A) Incident wave with simulation from pulse shaper
program B) Stress-strain curves
C) Calculated stress equilibrium R. D) Strain rate as function of specimen strain
Figure 6.14 Pre and post simulation of the test “Low strain rate” in section 6.6.
The high strain rate test shown in Figure 6.15 also shows a simulated Incident wave with higher
amplitude than the measured wave. In addition, the stress strain curves follow the same pattern as
the low strain rate case. The simulated stress-strain curve agrees well with the measured stress-
strain curve. The ripples at the beginning of the simulated stress-strain curve are due to the
simulated Incident wave. The stress rate changes in the beginning due to the elasto-plastic
behaviour of the pulse shaper. It takes several wave transits before the rate change is smoothed
out. Figure 6.15C shows that the stress equilibrium never reaches a steady state below 5% relative
deviation. The strain rate do not either find a steady plateau, but the pre-simulated, the measured,
and the post-simulated curve agrees in the evolution of strain rate (Figure 6.15D). Feeding the
measured Incident wave into the GW model generates the post-simulated curve.
0 50 100 150 2000
50
100
150
200
Time (s)
Stre
ss (M
Pa)
Incident Wave
Simulated Incident wave
0 1 2 3 40
50
100
150
200
Strain (%)
Stre
ss (M
Pa)
Test
Test - assumption of equil.
Pre simulation
0 1 2 3 40
5
10
15
20
25
30
35
40
Strain (%)
RE
Q (%
)
Pre simulation
Test
0 1 2 3 40
50
100
150
200
250
300
350
400
Strain (%)
Stra
in r
ate
(/s)
Pre simulation
Test
Post simulation
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A) Incident wave with simulation from pulse shaper
program
B) Stress strain curves
C) Stress Equilibrium D) Strain rate as function of specimen strain
Figure 6.15 Pre and post simulation of the test “High strain rate” in section 6.6
The coupling of the pulse-shaper simulation tool with the GW model creates a powerful tool to
design and validate incident waves. For example, first the pre-simulated pulse is created based on
the specimen design created with the design algorithm presented earlier. The Incident wave can
then be generated in the SHPB without the specimen, and the measured wave can be fed through
the GW model to simulate the response before an actual specimen is tested. This can help to
reduce the trial and error, which would be used to find an acceptable set-up.
6.9 Summary
A generalized wave mechanics model (GW model) was derived that describes the response of an
elastic specimen, dynamically loaded and situated between two elastic bodies. The model does not
assume bars with the same mechanical impedance and further the model do not have assumptions
of constant strain rate or stress equilibrium in the specimen. The model is then suitable to exploit
the strain rate and stress equilibrium of a test set-up.
The model was exploited to show that a linear rising pulse is the only shape that balances a
requirement of both constant strain rate and dynamic stress equilibrium. Next, the model was used
to calculate the number of wave transits to both dynamic equilibrium and constant strain rate for
any combination of machnical impedance of the specimen and bars. Further, the theoretical
maximum strain rate is independent of the specimen length and is instead controlled by the
impedance mismatches. Based on the algorithm to determine the equilibrium conditions, a design
algorithm was derived that calculated the maximum achievable strain rate for any combination of
specimen gage length and cross sectional area. The calculated setup satisfied both stress
0 50 100 1500
50
100
150
200
250
300
Time (s)
Stre
ss (M
Pa)
Incident Wave
Simulated Incident wave
0 1 2 3 40
50
100
150
200
Strain (%)
Stre
ss (M
Pa)
Test
Test - assumption of equil.
Pre simulation
0 1 2 3 40
5
10
15
20
25
30
35
40
Strain (%)
RE
Q (%
)
Pre simulation
Test
0 1 2 3 40
100
200
300
400
500
600
700
800
Strain (%)
Stra
in r
ate
(/s)
Pre simulation
Test
Post simulation
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equilibrium and constant strain rate at the same time. The GW model was implemented into a
simulation tool, which coupled simulation of Incident waves with post processing of the simulated
test. Together, the derived algorithms give a powerful tool to design tests in the SHPB and reduce
the trial and errors towards a successful set-up.
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7 An alternative momentum trap for the Split Hopkinson Pressure Bar
The Split Hopkinson Pressure Bar test rig is used for testing materials at high strain rates and
reveal their behaviour under dynamic loadings. The loading rate is controlled by shaping of the
Incident wave and the materials response is calculated from the Incident, Reflected and
Transmitted wave. However, the reflected wave goes forth and back in the Incident bar and loads
the specimen multiple times if the Transmitted bar does not move sufficient. This is the case in
most tests as the duration and amplitude of the Incident wave is designed to fracture the specimen,
which results in Reflected wave with longer duration and higher amplitude than the Transmitted
wave. These repeated loadings are unwanted if the fracture load has to be coupled with a post
analysis of the specimen, in for example a microscope.
To avoid the repeated loading the Incident bar has to be stopped after the first loading. The setup
for this is called momentum traps. Two types of momentum traps are found in literature. The first
setup was proposed by Nasser et al. [74] and is shown in Figure 7.1 The principle works with an
tube fitted around the Incident bar and an added flange to the Incident bar. A reaction mass rests
against the incident tube and restricts motion of the tube in the opposite end of the striker bar
impact end. The incident tube has the same length as the striker bar. A compressive wave was
generated in both the striker bar and the Incident bar upon impact. The wave in the tube travelled
to the reaction mass where it reflected as compressive again, since the reaction mass acted as a
stiff barrier. This compressive wave cames back to the flange when the tensile wave in the striker
bar reached the interface and unloaded the Incident bar. The compressive wave in the tube then
reflected off the flange and created a tensile wave in Incident bar. Thus, the incident compressive
wave was followed by a tensile wave, which reversed the movement of the Incident bar/specimen
interface after the Incident wave.
A) Principle of the momentum trap B) Picture of setup.
Figure 7.1 Momentum trap by Nemat Nasser [74]
The second type of momentum trap was proposed by Song [83]. It consists of a flange attached to
the impact end of the Incident bar and a reaction mass mounted around the Incident bar. The
distance between the flange and the reaction mass is set according to
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𝐷𝐼 = ∫ 𝐶𝐼𝜀𝐼(𝑡)𝑑𝑡𝜏
0
(7.1)
CI is the elastic wave speed in the Incident bar and 𝜀𝑖(𝑡) is the incident wave. When the flange
reach the reaction mass the incident cannot move further and repeated loading is avoided. Figure
7.2 shows a schematic drawing of the concept.
Figure 7.2 Momentum trap proposed by Song [83]
Pro and cons for the two methods has been listed in Table 7.1.
Table 7.1 Comparison of momentum traps
Pros Cons
Nasser et al. Work without precision pre-setting a distance in the setup
Incident tube must have same length as the striker bar.
Does not allow for controlled unloading of the
specimen.
The tube take some of the amplitude of the
incident wave and decrease the performance of the setup.
Requires special bearing for supporting the tube and the Incident bar has to rest in the tube.
Addition of reaction mass.
Song et al. Work for all sizes striker bars without exchanging any other parts
Allow for shaping of the unloading phase
Require precise distance setup
Requires incident pulses to be created with high
repeatability.
Addition of reaction mass.
The method by Song et al. requires less modification to the SHPB rig, but demands the
displacement of the Incident wave to be known precisely. Repeatability may be hard to obtain as
small variations in the gas gun pressure etc. creates different displacements from test to test under
the same settings. The method by Nasser et al. does not have this restriction, but the setup make
changes to striker bar length cumbersome as the tube length had to match the striker bar length.
Further, the tube reduces the momentum introduced in the system and the momentum transferred
to the specimen.
7.1 New method of momentum trapping
The distance between the bars was analysed to determine if multiple loading of the specimen did
occur for a given setup. The interface/bar interface displacements during the first loading is
calculated from the interface velocities as
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𝐷𝑎 = ∫ 𝐶𝐼(𝜀𝐼(𝜏) − 𝜀𝑅(𝜏))𝑑𝜏𝑡
0
(7.2)
𝐷𝑏 = ∫ 𝐶𝑇(𝜀𝑇(𝜏))𝑑𝜏𝑡
0
(7.3)
τ is the duration of the Incident wave. The final distance between the bars after the first loading
then become
𝐷𝑓𝑖𝑛𝑎𝑙 = 𝐿𝑠 − (𝐷𝑎 − 𝐷𝑏) (7.4)
After the Incident wave has compressed the specimen, the wave becomes the reflected wave,
which reverbed inside the Incident bar, and the Transmitted wave reverbed inside the Transmitter
bar. For simplicity it is assumed the bar ends is completely free after the specimen is broken. Thus
the reflection of the reflected wave becomes
𝜀𝑅𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑= −𝜀𝐼𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑
(7.5)
Then for each time the reflected wave passed interface (a) the interface moves the distance
𝐷𝑎(𝑛1) = 2 ∫ 𝐶𝐼 (𝜀𝐼𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑(𝜏)) 𝑑𝜏
𝑡
0
(7.6)
And accordingly for interface (b)
𝐷𝑏(𝑛2) = −2 ∫ 𝐶𝑇(𝜀𝑇𝑇𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑒𝑑(𝜏))𝑑𝜏
𝑡
0
(7.7)
The number of reverberation in the Incident bar is n1with a period of
𝑡𝐼0 =2𝐿𝐼
𝐶𝐼
(7.8)
For the Transmitter bar with reverberation number n2 the period is
𝑡𝑇0 =2𝐿𝑇
𝐶𝑇
(7.9)
The total displacement of interface (a) as
𝐷𝑎𝑡𝑜𝑡𝑎𝑙(𝑡) = 𝐷𝑎 + 𝑛1𝐷𝑎(𝑛1) + 2 ∫ 𝐶𝐼 (𝜀𝐼𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑(𝜏)) 𝑑𝜏
𝑡
𝑡−𝑛1∙𝑡𝐼0
, 𝑛1 = ⌊𝑡
𝑡𝐼0
⌋ (7.10)
For interface (b) the total displacement is
𝐷𝑏𝑡𝑜𝑡𝑎𝑙(𝑡) = 𝐷𝑏 + 𝑛2𝐷𝑎(𝑛1) + 2 ∫ 𝐶𝑇 (𝜀𝐼𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑(𝜏)) 𝑑𝜏
𝑡
𝑡−𝑛2∙𝑡𝑇0
, 𝑛2 = ⌊𝑡
𝑡𝑇0
⌋ (7.11)
By evaluating equations (7.10) and (7.11) for a certain time it could be observed if the distance
between the bars became smaller than the minimum distance obtained during the compression of
the specimen. If this was the case, the specimen was subjected to multiple loadings. Figure 7.3B
and C shows an example of the use of equations (7.10) and (7.11). The relative interface
displacement is calculated as (𝐷𝑎𝑡𝑜𝑡𝑎𝑙(𝑡) − 𝐷𝑏𝑡𝑜𝑡𝑎𝑙(𝑡)) − (𝐷𝑎 − 𝐷𝑏). Thus, Figure 7.3C gives the
relative motion of the interfaces after the first loading.
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A) Examples of waves. The
reflected and transmitted wave
reverbs around in the Incident bar
and Transmitter bar respectively.
B) Plot of the interface position for
a setup without multiple loading of
the specimen.
C) Relative interface movement.
Values below zero means the
interfaces approaches each other.
Figure 7.3 Interface motion and evaluation if multiple loading had happen.
Table 7.2 Details for setups in Figure 7.3C.
Setup Specimen details Incident bar Transmitter bar Striker bar
Setup with multiple
loading
15x15x15mm
(W x H x Ls)
Stainless steel
D = 25mm
L = 2001mm E = 191GPa
Ρ = 7741kg
Stainless steel
D = 25mm
L = 2000mm E = 191GPa
Ρ = 7741kg
D = 25mm
LST = 100mm V0 = 15.7m/s
Setup without multiple loading
12x12x10mm (W x H x Ls)
Stainless steel
L = 2001mm E = 191GPa
Ρ = 7741kg
Aluminium
L = 617 mm E = 73.5GPa
Ρ = 2809 kg
D = 25mm
LST = 150mm V0 = 19m/s
The setup with negative values of relative interface movement compressed the specimen further
than the initial loading. The exact numbers in Figure 7.3C here are not correct, as they were
calculated without the specimen between the bars after the initial loading. If the specimen are
present, the Reflected wave would lose energy every time the specimen was hit, while the
Transmitted wave would gain energy. Thus, the plot in Figure 7.3C only indicates if multiple
loading occurs or not.
Due to the short loading time, even for multiple loading, the specimen doesn’t have time to
accelerate downward, by the gravity, between the bars. If the bar diameter is 25mm and the
specimen would have to move 20mm or more to be completely free of the bars. Using constant
acceleration, this would take 64ms and within 64ms, the Incident bar would hit the specimen more
than 128 times according to the timescale in Figure 7.3C.
As indicated in Figure 7.3 an alternative way to avoid the specimen to be loaded multiple times is
to ensure the Transmitter bar moves faster away from the specimen than the Incident bar
approaches it after specimen failure. The displacement of the bar interfaces is controlled by the
displacement of the Reflected and Transmitted wave, and the frequency at which the wave reached
the interface. This frequency is controlled by the Transmitter bar length and the wave velocity of
the bar. Multiple loading can be avoided by designing the Transmitter bar to move faster away
from the specimen, by adjusting the displacement of the reverbing wave and the reverberation
frequency. Thus, by exchanging the Transmitter bar, to a bar of suitable length and material, single
loading of the specimen can be achieved.
In the SHPB setup described in chapter 6 the Transmitter bar were original a 2mtr steel bar, but
was exchanged to an aluminium bar to check the findings here, and the GW model was used to
0 200 400 600 800-1
-0.5
0
0.5
1
Time (s)
Ou
tpu
t vo
ltag
e (V
)
Incident bar
Transmitter bar
Incident waveTransmitted wave
Reflected wave
0 0.5 1 1.5 2 2.5 3 3.50
5
10
15
20
Time (ms)
Inte
rfac
e p
osi
tio
n (m
m)
Interface (a)
Interface (b)
Distance after the
incident wave has
passed.
0 1 2 3 4-1
0
1
2
3
4
Time (ms)
Rel
ativ
e in
terf
ace
mo
vem
ent
(mm
)
Setup without multiple loadings
Setup with multiple loadings
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simulate Transmitter bars made from from different materials. Details are given in Table 7.3
Table 7.3 Details for SHPB setup with modified Transmitter bar
Part Details
Specimen 12 x 12 x 10 mm
Es = 10GPa ρs =1953 kg/m3
Incident bar DI = 25mm
LI = 2001mm
EI = 191GPa ρI = 7741kg/m3
Transmitter bar
(To be validated)
DT = 25mm
LT = 617mm
ET = 73.5GPa ρT = 2809kg/m3
The Incident waves were identical to those presented in Figure 6.12 and is presented as low,
medium and high strain rate. The new Transmitter bar was made from aluminium and shortened to
617mm in length and was exchanged for a stainless steel bar similar to the Incident bar described
in Table 7.3. In principle, the Transmitter bar could be varied arbitrarily in size and material.
However, in reality the material choice is limited so the bar material alternatives was narrowed
down to an aluminium, a steel and a magnesium bar. Their diameters were locked to the diameter
of the Incident bar while the length could be varied as needed. The “low strain rate” Incident wave
was used to compare the materials. The GW model was set up with the steel Incident bar and the
Transmitted bar made from one of the materials to choose between. The Reflected and Transmitted
waves were calculated and equations (7.10) and (7.11) were combined to calculate the relative
distance between the bars after the first loading to failure. The calculations were carried out for
lengths between 0.1 and 3mtr of the Transmitter bar. If the relative distance was found to be
negative within 10 reverberations of the Reflected wave in the Incident bar, then the design was
found non useable. The choice of Transmitted bar influenced the obtainable strain rate and for
each material, the maximum strain rate was calculated.
The wavelength of the transmitted wave were also considered as short lengths of the Transmitter
bar increases the risk of overlapping waves at the strain gage. The wavelength was calculated by
multiplying the duration of the simulated Transmitted wave with the wave velocity of the bar
material. Table 7.4 present the findings of the simulations. The reflection coefficients B1 and B2,
and the maximum strain rate and the corresponding stress rate in the Incident bar are also included.
Table 7.4 Simulation results
Table header Aluminium Stainless steel Magnesium
E (GPa) 73.5 191 45
ρ (kg/m3)
2809 7741 1738
B1
(Reflection coefficient – Specimen to Incident bar)
0.932 0.932 0.932
B2 (Reflection coefficient – Specimen to Transmitter bar)
0.827 0.932 0.734
Max strain rate (/s) 219 121 303
Max stress rate (MPa/µs) 1.17 0.37 2.44
Calculated wave length (mm) 670 590 1020
Maximum bar length (mm) 1200 400 2000
Loading to specimen failure Yes Yes No
The simulation results in Table 7.4 shows that the stainless steel material requires a Transmitter
bar of maximum 0.4m to allow the bar to move away quickly enough. This is problematic as the
strain gage always should be placed 10 times the bar diameter away from the specimen end [18,
71]. The bar iss 25mm in diameter so it leaves 150mm bar behind the strain gage and a maximum
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wavelength of 300mm to avoid the wave overlapping at the strain gage. The calculated wavelength
for the Transmitted wave is 590mm so the wave will overlaps itself at the strain gage with a
Transmitter bar length of 400mm. Further, the maximum strain rate calculated for stress
equilibrium and constant strain rate at specimen fracture, is low due to a high impedance
mismatch, which requires many wave reverberations to ensure constant strain rate according to
Figure 6.10.
The magnesium bar gives a maximum length of 2mtr. Since magnesium is less stiff than steel, it
will deform more for the same load and move faster away than a steel bar of the same length.
However, the magnesium bar is not able to load the specimen up to fracture. The specimen would
simply push the bar away before fracture according to the simulation. The aluminium material
provided the compromise between the steel and magnesium options. The maximum bar length is
calculated to 1,2mtr and a wavelength to 670mm and an inability to fracture the specimen. Thus,
aluminium is validated as the material choice for the bar. The maximum bar length were then
calculated for the other Incident waves in Figure 6.12. The “medium rate” allowed a ma imum
length of 8 mm and the “high rate” allowed a ma imum length of 6 mm. Thus, the 617mm
aluminium bar is on the low end regarding the length. Data for tests performed with the bar with
all the Incident waves from Figure 6.12 were used for check if repeating loading has occurred. The
displacement of the Reflected and Incident wave were calculated from equations (7.6) and (7.7),
and the relative interface movement were calculated from equations (7.10) and (7.11). The
calculations are plotted in Figure 7.4 and they show that the “low rate” and “medium rate” tests
did not have repeated loading whereas the “high rate” had a small degree of repeated loading
(Relative interface movement below 0mm at a few times).
Figure 7.4 Relative interface movement for different
specimen strain rates
7.2 Summary
An alternative momentum trap method for the compressional SHPB was presented. The method
0 0.5 1 1.5 2 2.5 3-1
0
1
2
3
4
Time (ms)
Rel
ativ
e in
terf
ace
mo
vem
ent
(mm
)
Low strain rate
Medium strain rate
High Strain rate
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works by using a short and light Transmitter bar, such that it moves faster than the Incident bar
after specimen fracture. This ensures no further compression occurs on the specimen. A design
example was given in the form of a validity check of an already installed version of the method.
The design example included the necessary equations to calculate if the specimen experienced
repeated loading, and it was described how the GW model from chapter 6 were used to simulate
the design example. This proposed momentum trap method was simpler to apply as it only
requires an exchange of the Transmitter bar. However, the length and choice of bar material must
be selected with respect to the specimen to test, and the Incident waves to be applied. The bar will
be limited to these settings as a change in the Incident wave may change the ratio of the Incident
bar and Transmitter bar movement. A bar design is then designed to a given setup, and do not
work universal for all selections of Incident waves and specimen size and material.
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8 Through Thickness strain rate sensitivity of Eglass/Epoxy and Eglass/Lpet UD composite
This chapter present an investigation of the through thickness strain rate behaviour of two fibre-
reinforced polymers (FRP). An Eglass/Epoxy (Thermoset matrix) and an Eglass/Lpet
(Thermoplastic matrix) FRP is compression tested from 150 to 400/s with the SHPB described in
chapter 5. The results was used as validation cases for the model presented in chapter 6, and as
input data for the RESIST project mentioned in the preface.
8.1 Materials
Material specifications for the Eglass/Lpet and Eglass/Epoxy materials are shown in Table 8.1.
Specimens were cut with a diamond saw and milled to final dimensions of 12x12x10mm (width x
lenght x thickness). Examples of the specimens are shown in Figure 8.1. Department of Wind
Energy (DTU) tested the materials under static condition and all result noted as “static” comes
from [62, 63]. In the static case the specimen size was 20 x 20 x 40 mm (width x length x
thickness) for stiffness measurements and 25 x 25 x 40mm (w x h x t) with a waist cross section of
16 x 16mm (w x h), The latter was used for strength testing only.
Table 8.1 Material details
Property Eglass/Epoxy Eglass/LPET
Panel size 200x200x21,9mm 250x250x22mm
Fabrics Non crimp fabric SAERTEX
S14EU990-00940-T1300-499000
With 950g/m2
E-glass 1200 tex 4588
WG2-LPET-1000-UD
With 1050g/m2
Area weight 43,4kg/m2
Layup [0°]36 [0°]44
Matrix Araldite LY 564 Epoxy with Aradur 917 harder and DY 070 accelerator
LPET
Production method Vacuum Infusion Vacuum consolidation
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A) Specimens for high strain rate
testing
B) Specimens for quasi-static test
of strength [62, 63].
C) Specimens for quasi-static test
of the elastic modulus. [62, 63].
Figure 8.1 Test specimens
8.2 Test setup
The Split Hopkinson Pressure Bar at University of Southampton was used for tests at high strain
rates. The test rig is described in chapters 5, 6, and 7, and the test rig is shown in Figure 8.2
together with typical waves from a test. The test rig is equipped with a stainless steel Incident bar
and a stainless steel Transmitter bar, both 2mtr long, however the Transmitter bar is changed to a
0.617mtr long aluminium bar to ensure single loading of the specimens according to chapter 7.
A) The Split Hopkinson Pressure Bar B) Typical recorded waves from a test
Figure 8.2 The Split Hopkinson Pressure Bar
Pulse shaping with copper discs as described in chapter 5 was used to enable test at different strain
rates. Originally, the pulse shaping was done with a rounding of the impact end of the Incident bar,
however this was insufficient to produce the required Incident waves. The pulse shaper code
described in chapter 5 and the GW model described in chapter 6 was used to design three Incident
waves to achieve three strain rates. Examples of the waves are given in Figure 8.3B, and The
strain gage position is the distance between the strain gage on the bar and the specimen.
Table 8.3 gives details of set-up. Figure 8.3 also shows waves from the two pulse shaping
methods. The bar calibration method described in chapter 5 and details of the bars are provided in
table 8.2.
0 100 200 300-150
-100
-50
0
50
100
150
Time(s)
Stre
ss(M
Pa)
Incident wave
Reflected wave
Transmitted wave
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Table 8.2 Bar data
Parameter Incident bar Transmitter bar Striker bar
Material Stainless steel Alu 7075 T651 Stainless steel
Length (mm) 2001 617 100/150mm
Strain gage position (mm) 1021 276 -
E (GPa) 191 73.4 191
Density ρ kg/m3) 7741 2809 7741
Wave speed C ( m/s) 4978 5112 4978
The strain gage position is the distance between the strain gage on the bar and the specimen.
Table 8.3 Pulse shaper setup
Low Strain rate Medium strain rate High Strain rate
Material Copper Copper Copper
Pulse shaper dimension
(Diameter * Thickness) 8.73mm x 2mm 9.53mm x 2mm 11.41mm x 2mm
Strikerbar length (mm) 150 150 100
Impact velocity (m/s) 13 19 24
A) Rounded bar pulse shaping B) Copper disc pulse shaping
Figure 8.3 Incident waves from the Rounded bar pulse shaping method and the Copper dics method
The test rig were equipped with a gas gun able to achieve striker bar velocities up to 30m/s. The
velocity was measured with a light gage mounted in the muzzle of the gas gun. The bars were
equipped with two strain gages each. The strain gages had a gage length of 2mm and were placed
opposite to each other, and connected in bending compensation mode as shown in Figure 8.4B.
0 20 40 60 80 1000
100
200
300
400
500
Time (s)
Inci
den
t b
ar s
tres
s (M
Pa)
6.3m/s
15.6m/s
24.51m/s
0 50 100 150 200 250 3000
50
100
150
200
250
Time (s)
Inci
den
t b
ar s
tres
s (M
Pa)
Low strain rate
Medium strain rate
High strain rate
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A) Gage position on bar B) Wheatstone bridge setup
Figure 8.4 Strain gage setup on the bars
The strain gages were conditioned with a FYLDE HT379TA strain gage amplifier with a slew rate
of 8V/µs. The setup was shunt calibrated before performing a test. A four channel Picoscope 4427
scope was used for collecting data from the Incident and Transmitter bar and the light gage, at a
sampling rate of 20 MHz.
The collected data were post processed in a custom-made MatLab program to produce a stress-
strain curve and a strain rate curve for each test. The time synchronisation method described in
chapter 5 was used to synchronise the waves before calculating the stress strain curve. Since the
bars were unequal, the “modern” formulas, equations (8.1), were used for calculating stress and
strain.
𝜎(𝑎, 𝑡) =𝐴𝐼𝐸𝐼
𝐴𝑠
(𝜀𝐼(𝑡) + 𝜀𝑅(𝑡))
𝜎(𝑏, 𝑡) =𝐴𝑇𝐸𝑇
𝐴𝑠
(𝜀𝑇(𝑡))
𝜀(𝑡) =1
𝐿𝑠
∫ 𝐶𝐼(𝜀𝐼(𝜏) − 𝜀𝑅(𝜏)) − 𝐶𝑇𝜀𝑇(𝜏)𝑡
0
𝑑𝜏
(8.1)
Five specimens were tested for each strain rate and for each material, such that in total 30 tests
were conducted. The specimen dimensions were kept constant for all loading rates and were the
same for both the Eglass/Epoxy and Eglass/Lpet material.
8.3 Results
The failure stress and failure strain, and elastic modulus were calculated from the stress-strain
curve. The maximum stress defined the failure stress and the corresponding strain defined the
failure strain. The elastic modulus was calculated in the interval 1 – 1.8% strain. Strain rates were
defined as the average strain rate in the strain interval from 1 – 2% strain. The three strain rate
levels were named “ ow”, “ edium” and “High”. Typical profiles are shown in Figure 8.6A and
the obtained strain rates is shown in Figure 8.5B. Figure 8.6 shows the repeatability of the strain
rate profiles. The examination of the setup in chapter 6 proposed that constant strain could not be
reached with the high stress rate in the Incident wave for the given specimen/bar combination. The
medium and low strain rate showed a more stable evolution in strain rate. However, the low strain
rate showed a distinctive bend in the strain rate. Figure 8.6 shows it was present for all specimens
and of the two materials. The Eglass/Epoxy developed a higher strain before the acceleration in
strain rate compared to the Eglass/Lpet.
Strain gage
Strain gage
e-out
E-i
n
Dum
my
Dum
my
Gag
e 1
Gag
e 3
Wire connection to the bar
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A) Strain rate vs. strain. B) Strain rates shown with 95% confidence interval.
Figure 8.5 Strain rates
A) Eglass/Lpet – low strain rate B) Eglass/Epoxy – low strain rate
C) Eglass/Lpet – high strain rates D) Eglass/Epoxy – low strain rates
Figure 8.6 Strain rate vs. strain curves
Figure 8.7 shows stress strain curves for the low and high strain rate of the Eglass/Lpet material.
The high strain rate curves were affected by the noisy incident waves. The pulse shaper design
0 1 2 3 40
100
200
300
400
500
600
700
Strain(%)
Stra
inra
te (
/s)
Strain rates
Low strain rate
Medium strain rate
High strain rate
Strain rate
E/L
PET S
tatic
E/L
PET L
ow
E/L
PET M
ediu
m
E/L
PET H
igh
E/E
POXY s
tatic
E/E
poxy lo
w
E/E
poxy M
ediu
m
E/E
poxy H
igh
0
200
400
600
Str
ain
rate
(/s
)
-4 -3 -2 -1 0-700
-600
-500
-400
-300
-200
-100
0
Strain(%)
Stra
inra
te (
/s)
Strain rates
01
02
03
04
05
-4 -3 -2 -1 0-500
-400
-300
-200
-100
0
Strain(%)
Stra
inra
te (
/s)
Strain rates
25
02
03
04
05
-4 -3 -2 -1 0-700
-600
-500
-400
-300
-200
-100
0
Strain(%)
Stra
inra
te (
/s)
Strain rates
11
12
13
14
15
-4 -3 -2 -1 0-700
-600
-500
-400
-300
-200
-100
0
Strain(%)
Stra
inra
te (
/s)
Strain rates
11
12
13
14
15
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with a diameter of 11.4mm for this group was on the high limit according to the finding in chapter
5. The fact that the curves in Figure 8.7B shows unloading stages, reveals that the quality of the
generated Incident waves was doubtful.
A) Eglass/Lpet at low strain rate B) Eglass/Lpet at high strain rate
Figure 8.7 Examples of stress strain curves.
Table 8.5 shows the calculated mean value and 95% confidence interval in parentheses for the
failure stress, the failure strain, the E modulus, and the average obtained strain rate.
Table 8.4 Eglass/Lpet – TT Compression
Nominal strain rate
Property Unit Static 1/s 150/s 400/s
Failure Stress (X33t) MPa 120.4 (81.1) 166 (2) 170 (14) 189 (4)
Failure Strain (XS33t) % 1.64 (0.12) 2.40 (0.06) 2.20 (0.3) 1.90 (0.25)
Elastic modulus (E33t) GPa 11.1 (0.1) 8.7 (0.6) 7.1 (1.1) 9.0 (0.7)
Obtained strain rate /s 154 (7) 285 (37) 433 (38)
Table 8.5 Eglass/Epoxy – TT Compression
Nominal strain rate
Property Unit Static 1/s 150/s 400/s
Failure Stress (X11t) MPa 186 (15) 183 (10) 193 (11) 203 (12)
Failure Strain (XS11t) % 1.84 (0.23) 2.73 (0.15) 2.58 (0.41) 2.63 (0.55)
Elastic modulus (E33c) GPa 16.0 (0.7) 8.7 (0.9) 9.6 (1.0) 8.4 (2.6)
Obtained strain rate /s 262 (16) 302 (2) 487 (46)
Figure 8.8 shows the mean values plotted with their confidence interval. Eglass/Lpet showed first
an increase in failure strain and then a drop with increasing strain rate. This indicated a change in
failure mechanism with strain rate. Eglass/Epoxy showed an increase in failure strain from quasi
static to the lowest strain rate and then a stable level. Both materials experienced an increase in
failure stress with strain rate with the highest increase for the Eglass/Lpet. The Eglass/Lpet also
showed a higher increase in failure stress from quasi-static strain rate to the lower strain compared
to the Eglass/Epoxy.
-5 -4 -3 -2 -1 0-200
-150
-100
-50
0
Strain(%)
Stre
ss (M
Pa)
Stress strain curves
01
02
03
04
05
-4 -3 -2 -1 0-200
-150
-100
-50
0
Strain(%)
Stre
ss (M
Pa)
Stress strain curves
11
12
13
14
15
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Figure 8.8 Graphical representation of the results with confidence intervals
For both materials the elastic modulus decreased with increasing strain rate. Figure 8.9 shows
stress-strain curves of Eglass/Lpet obtained in the SHPB, and under quasi-static conditions. 90°
inplane compression tests was also carried out under quasi static conditions and for a UD lay-up,
the 90° direction should be the same as the through thickness direction. There is slight difference
in elastic modulus between the SHPB test and the quasi-static tests. Most of the decrease in the E
modulus comes from the difference in strain interval used for calculating the elastic modulus. In
the quasi-static test, the range 0.05-025% was used, whereas 1-1.8% was used for the SHPB tests.
The data below 1% strain were acquired in a non-equilibrium state.
Figure 8.9 Stress-strain curves for Eglass/Epoxy
Figure 8.10 shows photographs of the specimens after test. Despite the difference in geometry of
the quasi-static and the SHPB specimens, they possessed the similar crack patterns. The
Eglass/Lpet specimens seemed to possess more brittle behaviour at the higher strain rates with
smaller cracks. This finding was in agreement with the stress-strain curves, where a more brittle
behaviour was seen with increasing strain rate.
0 0.5 1 1.5 2 2.5 3 3.50
50
100
150
200
Strain (%)
Stre
ss (M
Pa)
Through Thickness - Compression - Static
90 - Compression - Static
SHPB - Compression
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Static
Low
Medium
High
A) Eglass/Epoxy B) Eglass/LPET
Figure 8.10 Post-test photographs of the specimens. Photographs of the static specimen were taken
from [62, 63].
8.4 Digital Image Correlation Check
Four tests of Eglass/Epoxy at low strain rate were conducted with a Vision Research V1610 high-
speed camera to monitor the SHPB test. A speckle pattern was painted on the surface of the
specimen and the Digital Image Correlation software ARAMIS 2D from GOM [99] was used to
determine the strain field on the surface. The strain measurements were taken as an average of the
facets within the red square in Figure 8.11A. The DIC strain measure was used with the stress
measure from the SHPB to generate stress strain curves. Comparisons of the elastic modulus are
shown in Table 8.6 and the stress strain curves are shown in Figure 8.11B. A paired T test was
conducted with the hypothesis, that the means were equal. The P value was found to be P = 0.49,
so the mean level were not significantly different. The DIC measurement then verified the use of
the “modern” post processing equations.
Table 8.6 Estimated elastic modulus
Specimen SHPB analysis
E modulus (GPa)
DIC Analysis
E modulus (GPa)
01 7.54 7.83
02 9.34 9.83
03 9.66 8.85
04 8.65 7.33
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A) Extraction of strain data. B) Comparison of stress strain curves.
Figure 8.11 DIC data post processing.
Figure 8.12 shows the how the major strain concentrates in a small region in the specimen for an
Eglass/Epoxy. The concentration developed already at 133MPa which corresponded to a average
major strain of 1.5%. This was the point where the strain rate increased from a stable level in the
low strain rate test, so the DIC measurement suggested that the reason for the bi linear evolution in
strain rate was due failure progression in the specimen.
Figure 8.12 Example of how the major strain evolves from an uniaxial state into a state with strain
concentration upon failure. At -133MPa it was seen that the strain started to concentrate at the right side at
the Transmitter bar.
8.4.1 Microscope investigation
Some specimens were post analysed by capturing scanning electron images of the fracture
surfaces. No differences were found between specimens tested at different strain rates. However,
there was a significant different is fracture surfaces. Figure 8.13 shows that the Eglass/Lpet had a
more ductile behaviour in terms of less splitting of the matrix compared to the Eglass/Epoxy,
which has little adherence of epoxy to the fibers. This is contradictory to the behaviour in the
failure strain plot where the Eglass/Epoxy had higher strain to failure for all strain rates. The
momentum trap installed in the test rig ensured that fracture surface was created with a single
loading.
-5 -4 -3 -2 -1 0-250
-200
-150
-100
-50
0Stress strain curve
Strain (%)
Stre
ss (M
Pa)
Stress Strain curve - SHPB Analysis
Stress Strain curve - DIC measurement
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A) Eglass/Lpet B) Eglass/Epoxy
Figure 8.13 Scanning electron microscope images.
8.5 Summary
A thermoset (Eglass/Epoxy) and thermoplastic (Eglass/LPET) fibre reinforced material was tested
for strain rate sensitivity in the thickness direction for a UD layup in a Split Hopkinson Pressure
Bar. The strain rates ranged from quasi static to 400/s.
The failure stress of Eglass/LPET showed a higher sensitivity to strain rate than Eglass/Epoxy. For
failure strain, the Eglass/Lpet showed an increase from quasi-static strain to 150/s and then a
slowly decreasing tendency with increasing strain rate. This behaviour indicates a shift in failure
mode inside the material.
For both materials, the elastic modulus was found to decrease with increasing strain rate.
DIC measurements suggested that the bilinear evolution in the strain rate was due to failure
progression in the specimen.
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9 Summary and Conclusion
The high strain rate characterisation of FRP materials present the experimenter with a new set of
challenges in obtaining valid data. These challenges were addressed in this work.
Chapter 1 introduced the concept of wave motion in solids, and the wave equation was used to
introduce the stress equilibrium process in a solid. Theoretical maximum impact velocities were
presented to show that the impact velocity could not be endlessly increased to increase the strain
rate. The approach also indicated that the applied deformation history should be controlled and
adapted to the test material to reach a homogenous stress state in the specimen before failure. The
high-speed servo-hydraulic test machine and the Split Hopkinson Pressure Bar are used for high
strain testing of FRP materials. They complement each other in their accessible strain rates as they
represented two distinct ways of introducing the deformation into the specimen. The methods were
widely used for testing metallic materials, but accepted testing standards may not be directly
transferrable to testing FRP materials. A literature survey indicated contradictory results that
possibly arise from challenges with the testing methods. The remaining work was devoted to re-
examining the high-speed servo-hydraulic test machine and the Split Hopkinson Pressure Bar in
terms of testing linear elastic FRP materials.
Chapter 2 presented a model to simulate the influence of the load train of a high-speed servo-
hydraulic test machine on the stress and deformation state in the test specimen. It was found that:
A linear elastic specimen cannot be deformed with a constant velocity in a high-speed
servo-hydraulic test machine due to inertial damping in the load train and open loop
control of the piston rod velocity.
The specimen deformation rate was controlled by the mass of the moving grip.
Load ringing occurred as an effect of high acceleration of the system and the amplitude
is determined by the acceleration of the system. The strategy with a built-in load cell
could not easily be realised for polymeric fibre composites as the measurement relies on
a complete linear response as well as the ability to vary the cross section of the
specimen.
The non-rigid connection between specimen and piston rod in the conventional slack
adapter led to oscillations in the velocity. The frequency of the oscillations was low
such that the specimen would fail within the first period of the oscilations.
The rigid connection created with the Fast Jaw system and combined with the fast
gripping mechanism, lead to oscillations in the load cell before fracture of the specimen.
A time lack between strain and stress was observed in the simulations at higher impact
velocities, which would influence the estimation of the E modulus.
Trapped air in a conventional slack adapter could cause high loading on the specimen
before high deformation velocities were reached. This could cause weak specimens to
fracture before a high strain rate was reached.
As far as specimen fabrication and load ringing are concerned, a conventional slack
adapter design may be preferable for testing elastic materials, whereas the Fast Jaw
system using the developed techniques is more suitable for testing metallic materials.
Chapter 3 presented the design and construction of a high-speed servo-hydraulic test machine
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The design process illustrated the added complexity and cost added to the test machine
to reach higher deformation velocities.
The grips and slack adapter were custom designed for the machine.
The final machine was able to obtain piston rod velocities in excess of 25m/s.
Chapter 4 presented material tests carried out on the test machine described in chapter 3.
An Eglass/Epoxy, an Eglass/Lpet and Carbon/PA6 material were tested at three impact
velocities and in three specimen configurations. In total more than 120 tests were
carried out. The layups were UD 0°, UD 90° and Quattro axial.
It was not possible to obtain constant strain rates for any of the tests.
It was not possible to test the 90° specimens at high strain rate in tension as they
fractured before they were accelerated to higher deformation velocities.
The general tendency was an increase in failure stress and failure strain.
The lastic modulus and Poisson’s ratio were in general unaffected by the increased
strain rate.
The test series presented certain difficulties, as it was not possible to obtain constant strain rates
for any of the tests. The model in chapter 2 was developed during the post-processing of the data
and the model supported the observed tendency, namely that a constant strain rate was not
possible. The model indicated the reason to be the inertial damping in the load train and the
inability of the system to maintain the impact velocity. Chapters 2 and 4 thus support each other in
the conclusion that it may be difficult to obtain constant strain rates in the high-speed servo-
hydraulic test machine. Furthermore, it may also be problematic to obtain even modest
deformation velocities for materials with low strain-to-failure.
Chapter 5 presented a description of the compressional Split Hopkinson Pressure Bar
It was found that the dominant pulse shaping method was limited by frictional
properties. The method thus possesses a limit in the stress rate that can be applied to the
test specimen.
The importance of accurate calibration of the test rig was emphasized and a simple new
method for calibrating the wave velocity was presented.
Chapter 6 presented a wave mechanics model (GW model) to estimate the specimen behaviour in
the SHPB test rig.
The model was derived without assumptions of either constant strain rate or stress
equilibrium and had restrictions on the choice of bars.
The literature had shown that the linear rising Incident wave would promote stress
equilibrium and constant strain rate. The model was used to predict the number of wave
transits in the specimen to achieve both stress equilibrium and constant strain rate for all
possible combinations of bars for a linear rising Incident wave. The findings provide
useful information for selecting bars and set-up. For example to model showed that a
high impedance mismatch at the Transmitter bar would facilitate stress equilibrium, but
extend the time to constant strain rate, unless the Incident bar had a low impedance
mismatch to the specimen.
The model was implemented into a design algorithm to design specimens for the SHPB
test rig. It was possible to estimate the maximum strain rate for a given specimen size
and the associated maximum stress rate in the Incident pulse.
It could be concluded that only linear rising Incident waves would satisfy both a stress
equilibrium state and a constant strain rate at the same time.
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The requirement of a linear rising pulse means the acceleration of the Incident bar must
be constant and not the velocity of the Incident bar. A constant velocity in the Incident
bar would result in a decreasing strain rate in the specimen throughout the test.
Chapter 7 presented a new momentum trap for the SHPB test rig.
The method works by modifying the Transmitter bar such that it would move faster
away from the specimen than the Incident bar would approach the specimen.
The consequences for stress equilibrium and strain could be assessed with the model in
chapter 6.
The method is easy to implement, but was restricted to the specimen set-up for which it
was designed.
The method was verified with both high-speed videos and the model from chapter 6.
Chapter 8 presented a set of compression tests in thickness direction of the materials in chapter 4
The pulse shaping method presented in chapter 5 was used to create smooth rising
Incident waves for three strain rates.
The momentum trap method described in chapter 7 was used to ensure single loading of
the specimens.
Strain measures were compared with DIC measurements. They agreed and validated the
set-up and the post-processing of the data.
The model in chapter 6 showed that the test fastest test were fracturing the specimen
before a constant strain rate was obtained. The measured data provided the same
information and helped validate the model in chapter 6.
An increase was seen in the failure stress for both the Eglass/Epoxy and the
Eglass/Lpet.
For the failure strain, there was a significant drop at the highest strain rate for the
Eglass/Lpet, whereas this drop was not seen with the Eglass/Epoxy.
to high strain rates
9.1 Perspective
When the findings are combined, the high-speed servo-hydraulic test machine is clearly less
suitable for high strain rate testing of FRP materials if an constant strain rate is desired. The set-up
with grips and slack adapter was found troublesome. In addition, the inertia forces proved to be
particularly difficult to handle. This is problematic because the high-speed servo-hydraulic test
machine could close the gap, in accessible strain rates, between quasi-static rates and the lower
strain rate limit of the SHPB. The SHPB was found to have the potential to achieve both constant
strain rates, and stress equilibrium. However, challenges exist in controlling the Incident waves,
which is the key to successful testing in the SHPB test rig. The developed model of the specimen
response would also be applicable for a tensile set-up, which would be the natural place to
continue this work. The tensile set-up has some extra challenges such as fixing the specimens to
the bars. Grips could corrupt the reflection point and would require the strain to be measured
directly on the specimen, for example with DIC. This also proved to be successful in this work.
Rasmus Normann Wilken Eriksen. March 2014.
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List of figures
Figure 1.1 Dynamics aspects of mechanical testing [4] ..................................................................................... 2 Figure 1.2 Bar deformed from initial length L to final length Lf with the increment dL ................................... 4 Figure 1.3 Strain measures and relative deviation compared to the engineering strain measure. ....................... 5 Figure 1.4 Calculated strain rate as function of the applied deformation ........................................................... 6 Figure 1.5 Deformation of bar – distribution of wave deformation ................................................................... 7 Figure 1.6 Interface between two slender bars ................................................................................................... 8 Figure 1.7 Simulation with LS Dyna of a 0.2m long and 0.02m wide bar. The nodes are rigidly clamped in
one end and in the other end the nodes are given an initial velocity of 20m/s which is maintained through how
the entire simulation. E = 1GPa, ρ=1000kg/m3. Poisson’s ratio is set to zero to remove dispersion noise from
the plot. .............................................................................................................................................................. 9 Figure 1.8 Maximum impact velocity for FRP material systems. X-axis is material systems. ........................ 10 Figure 1.9 Comparison of fibre bundle tensile strength (T-700 = Carbon fibre) [29] ...................................... 13 Figure 1.10 Tensile strength and failure strain for Aramid and polyethylene fibres (SK66 and UD66 ) [31].. 13 Figure . Schematic overview of the servo hydraulic test machine [ ]. The addition of a “ ost motion unit”
in the load train is a major different to standard one axis hydraulic test machines. The lost motion unit allows
the piston rod to accelerate to a given velocity before the specimen is loaded. ............................................... 15 Figure 2.2 Pressure drop at valve during opening and at steady piston rod velocity. Pmax/2 is lost when oil
passes the valve into the cylinder and a pressure of Pmax/2 is build up in the cylinder as the oil has to pass out
of the valve. 𝑥 is the piston rod acceleration and 𝑥 is the piston rod velocity. ................................................. 16 Figure 2.3 Different concepts for the slack adapter [12] .................................................................................. 17 Figure 2.4 Specimen design used with approach (c) in Figure 2.3. .................................................................. 18 Figure 2.5 Examples of ringing and guideline for acceptance of ringing. ........................................................ 18 Figure 2.6 Schematic overview of slack adapters. Note the slack adapter is running inside the piston rod for
the conventional slack adapter. ........................................................................................................................ 20 Figure 2.7 Simulation of the conventional slack adapter. ................................................................................ 24 Figure 2.8 Schematic specimen design ............................................................................................................ 24 Figure 2.9 Simulation and test of linear elastic material of straight side coupon specimen with cross section
=15mm2, Esp = 38GPa and Lg =50mm. .......................................................................................................... 25 Figure 2.10 Comparison between simulation and real test with air model included. The pressure was added as
pressure on the rubber and on the piston rod. .................................................................................................. 26 Figure 2.11 Specimen design used in the simulations of the Fast Jaws ........................................................... 27 Figure 2.12 Simulation of stress strain curves for piston rod velocity of 20 m/s ............................................. 28 Figure 2.13 Strain rate and force in the system. ............................................................................................... 28 Figure 2.14 The bouncing effect because of no rigid connection between the slack adapter and the specimen.
In figure B additional damping was added by inclusion of air in the slack adapter. ........................................ 29 Figure 2.15 Parameter study of classical slack adapter setup. The largest effect on the strain rates are found
from the specimen length and cross sectional area of the slack adapter. Impact velocity 20 m/s. ................... 31 Figure 2.16 Parameter study of classical slack adapter setup. Impact velocity 2 m/s. ..................................... 32 Figure 2.17Parameter study for elastic plastic specimen. Impact velocity 20m/s ............................................ 33 Figure 3.1 Design modules for design and construction of the HS machine. Illustration from [1]. ................. 35 Figure 3.2 Velocity profile of the piston rod as function of the displacement. M = 55kg. Ap= 4778mm2. Pn =
280 bar. ............................................................................................................................................................ 37 Figure 3.3 Valve selection chart – xacc is the acceleration length to full flow rate. Ap = 4778mm2 ................. 38 Figure 3.4 Frame available to build a high speed servo hydraulic test machine .............................................. 39 Figure 3.5 3D model of the test machine ......................................................................................................... 40 Figure 3.6 Installation of the test machine ....................................................................................................... 40 Figure 3.7 Trigger setup ................................................................................................................................... 42 Figure 3.8 Velocity calibration and used valve signal ..................................................................................... 42 Figure 3.9 Control electronics .......................................................................................................................... 43 Figure 3.10 Design of specimen and tabs ........................................................................................................ 44 Figure 3.11 Examples of tab failures ............................................................................................................... 45
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Figure 3.12 The upper grip system and the finite element simulation of the system ....................................... 46 Figure 3.13 Upper grip assembly and the mounting to the crosshead .............................................................. 46 Figure 3.14 Parts for the load train. The upper grip assembly is not shown. ................................................... 47 Figure 3.15 Impact damper for deaccleration of the lower grip. ...................................................................... 48 Figure 4.1 General specimen design ................................................................................................................ 50 Figure 4.2 Specimens before test for an impact velocity of 20 m/s.................................................................. 50 Figure 4.3 Examples of speckle patterns. ......................................................................................................... 51 Figure 4.4 Specimens after test for an impact velocity of 20 m/s .................................................................... 52 Figure 4.5 Example of stress strain curves for Eglass/Epoxy 0° loading direction .......................................... 53 Figure 4.6 Example of strain - strain rate curves. The non-constant strain rate was found for all tests! .......... 54 Figure 4.7 Averaged normalised Failure stress, Failure strain, Elastic modulus and poisons ratio as function of
strain rate ......................................................................................................................................................... 55 Figure 4.8 Example of curve from which poisons ratio is estimated as the slope of the curve. This curve is for
UD 0° Eglass/Epoxy at 20m/s. ......................................................................................................................... 56 Figure 4.9 Comparison of non-constant strain rate problem with results from Fitoussi et al. [57]. ................. 57 Figure 5.1 Schematic overview of the compressive Split Hopkinson Pressure Bar[69] .................................. 58 Figure 5.2 Example of a Split Hopkinson Pressure Bar at the University of Southampton ............................. 59 Figure 5.3 Examples of stress waves generated in the SHPB .......................................................................... 60 Figure 5.4 Schematic of the specimen situated between the Incident and Transmitter bars ............................. 61 Figure 5.5 The classic and modern SHPB setup. Subscript I and T in the modern SHPB indicate the
individual properties of the bars. ...................................................................................................................... 64 Figure 5.6 Pulse shaping technique – Conical striker bar ................................................................................ 64 Figure 5.7 Pulse shaping method with a cushion material. .............................................................................. 65 Figure 5.8 Schematic of the copper disc placed at the Incident bar with the interface velocities VPS1 and VPS2.
......................................................................................................................................................................... 65 Figure 5.9 Copper discs and available strikers. The Teflon coating shown together with the Striker bars, was
used for lubricating the barrel of the gas gun and is. Lubrication, as WD40, is not preferred as the tight
tolerances in the barrel causes the oil to pile up in front of the Striker bar and disturbs the impact................. 67 Figure 5.10 Pulse shaping with a copper disc .................................................................................................. 67 Figure 5.11 Parameter study of the four parameters. The basic set of parameters are the striker bar velocity
V0, the striker bar length, the pulse shaper diameter and the pulse shaper thickness. ...................................... 68 Figure 5.12 Various diameters (D) of 2mm thick copper disc tested under similar conditions. ....................... 69 Figure 5.13 Investigation of “ripples”. Pulse shapers were tested under various conditions to narrow down the
controlling parameters of ripples. .................................................................................................................... 69 Figure 5.14 LS Dyna simulation of pulse shapers. hp= 1mm, V0 = 16m/s, LST = 400mm [86]. ....................... 70 Figure 5.15 Effect of the accuracy of the elastic wave velocity C0 on the stress strain curve [89]. ................. 71 Figure 5.16The incident and reflected wave in the Incident bar and the error in estimating the elastic wave
velocity ............................................................................................................................................................ 72 Figure 5.17 Schematic of the wave traveling inside the Incident bar. The striker bar impact creates a
compressive wave, which travels in the positive direction. As the wave reflects at the other end it return as
tensile wave traveling in the negative direction. .............................................................................................. 72 Figure 5.18 A signal from the Incident bar along with its representation in the frequency domain. ................ 73 Figure 5.19 Synchronisation of the Incident, Reflected and Transmitted wave. .............................................. 75 Figure 5.20 Exact Solution of the first 3 vibrational modes to the Pochhammer-Chree equation for bar
material with ν . . Mode 1 is dominant in SHPB tests [72]. ......................................................................... 76 Figure 5.21 Initial sharp wave subjected to dispersion .................................................................................... 77 Figure 5.22 Incident waves and their associated frequency content................................................................. 77 Figure 5.23 Variation of the phase velocity as function of the normalised wave number. ............................... 78 Figure 5.24 Maximum frequency present in the incident wave before dispersion correction is required. v =
0.3, C0= 5000m/s. ............................................................................................................................................ 79 Figure 6.1 Schematic overview of the specimen and bar interfaces. The interface between the Incident bar and
the specimen is denoted (a), while the interface between the specimen and the Transmitter bar is denoted (b).
A is the cross sectional area. E is the elastic modulus, and ρ is the density. x defines the positive propagation
direction of waves in the specimen. ................................................................................................................. 82 Figure 6.2 Diagram for wave motion the specimen. 𝜎𝑛𝑅 is the reflected part of the wave at each reflection
between the wave and the interfaces. ............................................................................................................... 84
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Figure 6.3 Wave directions. ............................................................................................................................. 87 Figure 6.4 Comparison of the wave mechanics model (Equations 9 and 10 in [21]) presented by Frew et al.
and the GW model derived in this work (Equations (6.26) and (6.27)) ........................................................... 89 Figure 6.5 Comparison between the GW model and the formulas derived by Yang et al. [19]. ...................... 89 Figure 6.6 Different loading pulses .................................................................................................................. 90 Figure 6.7 Simulation of stress equilibrium and strain rate evolution for B1 = B2 = 0.667 ............................. 91 Figure 6.8 Simulation of stress equilibrium and strain rate evolution for B1 = B2 = 1. (Rigid boundaries) .... 91 Figure 6.9 Simulation of stress equilibrium and strain rate evolution for B1 = 0.667, B2 = 0.22. ................... 91 Figure 6.10 Number of reflections in the specimen to achieve stress equilibrium and constant strain rate ..... 94 Figure 6.11 Maximum strain rate and stress rate ............................................................................................. 96 Figure 6.12 Incident waves and strain rate vs. strain curves for the design case .............................................. 97 Figure 6.13 Schematic of the simulation model ............................................................................................... 98 Figure 6. Pre and post simulation of the test “ ow strain rate” in section 6.6. ............................................. 99 Figure 6. Pre and post simulation of the test “High strain rate” in section 6.6 ........................................... 100 Figure 7.1 Momentum trap by Nemat Nasser [74]......................................................................................... 102 Figure 7.2 Momentum trap proposed by Song [83] ....................................................................................... 103 Figure 7.3 Interface motion and evaluation if multiple loading had happen. ................................................. 105 Figure 7.4 Relative interface movement for different specimen strain rates .................................................. 107 Figure 8.1 Test specimens .............................................................................................................................. 110 Figure 8.2 The Split Hopkinson Pressure Bar ................................................................................................ 110 Figure 8.3 Incident waves from the Rounded bar pulse shaping method and the Copper dics method.......... 111 Figure 8.4 Strain gage setup on the bars ........................................................................................................ 112 Figure 8.5 Strain rates .................................................................................................................................... 113 Figure 8.6 Strain rate vs. strain curves ........................................................................................................... 113 Figure 8.7 Examples of stress strain curves. .................................................................................................. 114 Figure 8.8 Graphical representation of the results with confidence intervals ................................................. 115 Figure 8.9 Stress-strain curves for Eglass/Epoxy ........................................................................................... 115 Figure 8.10 Post-test photographs of the specimens. Photographs of the static specimen were taken from [62,
63]. ................................................................................................................................................................. 116 Figure 8.11 DIC data post processing. ........................................................................................................... 117 Figure 8.12 Example of how the major strain evolves from an uniaxial state into a state with strain
concentration upon failure. At -133MPa it was seen that the strain started to concentrate at the right side at the
Transmitter bar. .............................................................................................................................................. 117 Figure 8.13 Scanning electron microscope images. ....................................................................................... 118 Figure 9.1 State of the art single sensor cameras ........................................................................................... 127 Figure 9.2 Recording length and frame rate as function of the DIC noise floor. The DIC noise floor can also
be seen as a selected average strain increment over the specimen between each image. The user can select that
there should be x% strain increase between each image and this will set up the requirement for the camera.
....................................................................................................................................................................... 130 Figure 9.3 Interface between two slender bars ............................................................................................... 131 Figure 9.4 Reflection and transmission coefficients as functions the relative mechanical impedance and the
relative cross sectional area............................................................................................................................ 132 Figure 9.5 Sketch of cylinder with a piston which moves forward at a velocity v. As the piston moves forward
the air is compressed and forced out of the opening. ..................................................................................... 133 Figure 9.6Mass discharge rate calculated from equation (9.9) and (9.10) respectivily as function of the
pressure ratio. Increasing P1 with P2 constant is the situation with the cylinder compression the air.
Decreasing P2 with P1 constant is the situation where the ambient pressure is lowered and the cylinder
pressure is kept constant. ............................................................................................................................... 134 Figure 9.7 Simulations with A1 = 1283mm2, A2 = 18.6mm2, λ=1.4 (Air), M = 0.029kg/mol, R 8.315
j/mol*k, T = 293K (Assuming constant temperature), Chamber length = 0.23m. ......................................... 136 Figure 9.8 Non-reversible model to simulate a elasto-plastic material during a loading profile with unloading
stages. ............................................................................................................................................................ 137 Figure 9.9 Strain profile and stress strain curve of the non-reversible material model. ................................. 138 Figure 9.10 Selection curves for high strain rate strain gage amplifiers. ....................................................... 140
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List of tables
Table 1.1 Strain rates and different testing techniques adopted from [14] ......................................................... 2 Table 1.2 Formulas for average strain measures after deformation dL ................................................ 4 Table 1.3 Comparison of maximum impact velocity for steel and aluminium with equal yield
strength 9 Table 1.4 Collected material data from literature ............................................................................................. 11 Table 2.1 Mass distribution and equation systems for the slack adapter systems ............................... 21 Table 2.2 Specimen details for comparison tests ............................................................................... 24 Table 2.3 Specimen parameters and their nominal values ................................................................. 27 Table 2.4 Parameter description and their nominal values. ................................................................ 30 Table 3.1 Performance requirements ................................................................................................. 36 Table 3.2 Simplified firing sequence. When the user asked to fire, step 5 allowed the used to abort
the operation. 41 Table 4.1 Test matrix. IP = In plane. ................................................................................................. 49 Table 4.2 Test coupons structure and materials ................................................................................. 50 Table 4.3 Digital image correlation setup .......................................................................................... 51 Table 4.4 Parameter estimation .......................................................................................................... 52 Table 4.5 Failure stress (MPa) ........................................................................................................... 53 Table 4.6 Failure strain (%) ............................................................................................................... 53 Table 4.7 Elastic modulus (GPa) ....................................................................................................... 53 Table 4.8 Poisson’s ratio -) ................................................................................................................ 54 Table 4.9 Average strain rate (/s) ....................................................................................................... 54 Table 5.1 Formulas to post process SHPB data – Table partly adopted from Zhang et al. [73] ......... 63 Table 5.2 Calibration values for the incident and Transmitter bar. .................................................... 74 Table 6.1 Maximum strain rate criteria for linear elastic brittle specimens and equal bars................ 81 Table 6.2 Sequences for exponents in equation (6.24) for n=0 to 10 .................................................. 86 Table 6.3 SHPB specimen design algorithm ....................................................................................... 95 Table 6.4 Stress rates in Incident pulses in Figure 6.12 ...................................................................... 97 Table 7.1 Comparison of momentum traps ....................................................................................... 103 Table 7.2 Details for setups in Figure 7. . “Thk” is the compression dimension. ......................... 105 Table 7.3 Details for validation case. ............................................................................................... 106 Table 7.4 Simulation results ............................................................................................................ 106 Table 8.1 Material details ................................................................................................................ 109 Table 8.2 Bar data ............................................................................................................................ 111 Table 8.3 Pulse shaper setup ............................................................................................................ 111 Table 8.4 Eglass/Lpet – TT Compression ......................................................................................... 114 Table 8.5 Eglass/Epoxy – TT Compression ...................................................................................... 114 Table 8.6 Comparison of estimated elastic modulus ........................................................................ 116
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Appendix A High speed imaging for material testing
In high strain rate testing the test event may happen so fast that it is impossible to obtain any
qualitative information about the material test by observing the test or recording the event with a
standard camera at a standard frame rate (20-30fps). High-speed cameras can be employed to
obtain qualitative information from the event by capturing images at several thousand images per
second. The qualitative information can be used to assess the test in conjunction with load and
strain data and look for failure mechanism etc. The images can be turned into quantitative
information by using the images for Digital Image Correlation (DIC) and support or replace the
deformation and strain measurements. DIC is a well-known non-contact measurement technique,
which has been describe in great details in literature [100-102], further the use of high speed
imagining and DIC has also been described be several authors [103-106].
High speed cameras can be divided into 3 distinct types of cameras, the rotating mirror multiple
sensor camera, the single sensor and in situ storage single sensor cameras, but for DIC only the
two latter are interesting as the rotating mirrors causes many problems for DIC algorithms [107].
Examples of state of the art single sensor cameras are given in Figure 9.1 along with their
maximum frame rate at maximum resolution.
A) Photron SA Z. Single sensor
camera. 1024x1024px at max 20000
FPS.
B) Specialised Imaging – Kirana
single sensor in situ storage camera.
924x768px at max 5000000 FPS.
Figure 9.1 State of the art single sensor cameras
The main difference between the two types of cameras is that the single sensor camera will have
reduced resolution at higher frame rate than the maximum frame rate at maximum resolution. The
bottleneck is the data transfer rate from the sensor to the ring buffer memory. The single sensor in
situ camera comes around the transfer bottleneck by having built in ring buffer memory in the
sensor itself so the resolution can be kept high at even very high frame rates. The drawback is a
limited ring buffer memory. The Kirana camera can only store 180 images in its sensor memory
whereas the Photron SA Z can store several hundred thousand of images in its external RAM
memory. For FRP materials where the strain to fracture is low the few images may be enough to
cover the test event, however the triggering of the camera becomes challenging as For example the
recording time for the Kirana is 36µs at 5000000FPS.
In a situation of selecting a high speed camera for both qualitative and quantities image capturing a
lot of technical parameters must be selected as a compromise where the most important.
Resolution
Frame rate (FR)
Recording time
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Magnification in px/mm (M)
Field of view (FOV)
Shutter time (SS)
Frame rate, resolution, recording time and shutter time are pure camera properties whereas FOV
and M is a function of the physical setup with respect to the test subject.
The easiest parameter to select is the shutter time as is always should be minimised to minimise
blur in the images and in most circumstances the shutter time must be less than 1/FR and not 1/FR
which is the maximum shutter time. However the shutter time can be restricted by the available
light sources and the expected blur for the setup should always be evaluated. The maximum blur in
a scene can be calculated as a function of the maximum velocity (V) in the scene and the FOV and
M as
𝑚𝑏 = 𝑆𝑆 ∗ 𝑉 ∗ 𝑀 (9.1)
For single sensor cameras the resolution is limited at higher frame rates due to the transfer
bottleneck and this will also affect the magnification. For qualitative measurement a feature should
be occupied by at least two pixels [108] but for materials test of especially F P’s at least 6 -128
pixels should cover the shortest distance of the FOC to get a minimum overview of the fracture
process. The FOV is adjusted on the base of the resolution by the choice of lens and working
distance (Distance between specimen and sensor) such that the entire specimen is within the FOV
and covered by the required amount of pixels. The magnification is then the product of the
selected/available resolution to the cover the specimen.
The recording length should be long enough to cover the entire deformation process and for single
sensor cameras this is no problem as normally are equipped with several gigabyte of ring buffer
memory, but attention must be paid to the recording length of single sensor in situ cameras. The
duration of the specimen deformation can be estimated from the failure strain and the strain rate,
but it must be taken into account that the strain rate is not constant during the test and in most
circumstances are lower than predicted for the test setup.
For qualitative imaging there exist guidelines for selecting frame rates where For example simple
statements as a minimum of 10 images must be capture of the event [108] and required frame rate
can be calculated if the deformation duration is known. For quantitative measurement with the
images set of guidelines for selecting frame rate and recording duration can be derived based on
the properties of the accuracy of the non-contact measurement method. In many cases the selection
for frame rate and recording length may also determine the type of camera. Any measurement
method is limited by its noise floor For example how small a quantity can be measured which with
a reasonable certainty can be regarded as not being noise from the measurement method itself. In a
tensile test of a test specimen one end of the specimen will be clamped and the other will be
moving and thus there will be a velocity field over the specimen range from zero velocity at the
clamped end to maximum velocity at the moving end. The moving end will also experience the
largest displacement meaning that in terms of displacement the signal and signal to noise ratio will
be much larger at the moving end. Thus each end of the specimen will set different requirements
for measurement method if displacement is used as base; however the strain field is uniform over
the entire specimen and will have the same signal to noise ratio over the entire specimen. The
noise floor in DIC strain measurements is determined by a long range of parameters, which to
some degree can be controlled in the lab, however in the end the noise floor will have a certain
level and this level can be used to determine a set of requirement for the high speed cameras used
to capture images for DIC. For the commercial DIC system Aramis the general strain resolution
(εRES) is about 0.01% for a good setup. If the fracture strain (εf) in a UD loading situation of a
material is known then recording length in frames which covers the deformation event can be
determined. The strain change from frame to frame should not be less than the noise floor else the
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event is oversampled. Thus the feasible number of images (NoI) becomes
𝑁𝑜𝐼 =𝜀𝑓
𝜀𝑅𝐸𝑆
(9.2)
The number of images is plotted in Figure 9.2as function of the failure strain for different DIC
noise floor levels. The time to fracture (T) for an average strain rate 𝜀̇ yields
𝑇 =𝜀𝑓
𝜀̇ (9.3)
The required frame rate to obtain a feasible number of images up too fracture then becomes
𝑓𝑝𝑠 =𝑁𝑜𝐼
𝑇=
𝜀𝑓
𝜀𝑅𝐸𝑆
𝜀̇
𝜀𝑓
=𝜀̇
𝜀𝑅𝐸𝑆
(9.4)
The required frame rate is then only a function of the DIC noise floor. Thus the camera frame rate
and camera recording length can be selected as function of the inter frame strain increment which
should be set to or higher than the DIC noise floor. It should be noted that a higher resolution will
allow the user to use a higher facet size and step size without scarifying spatial resolution and a
reduced noise floor can be achieved. Thus the inter frame strain increment can be chosen to a
lower value and the effect is that high resolution camera will require higher frame rate and
recording length. Figure 9.2 shows graphs for selecting recording length and frame rate as function
of average strain rate and resolution of the DIC measurements. An example of selection could be
than a material with a technical fracture strain of 3% should be tested at an average strain rate of
1000 /s with DIC resolution/inter frame strain increment of 0.02%. This would call for a recording
length of about 150 images and a frame rate of about 5e6 fps. There should be left room enough
for a lower strain rate with respect to the recording length so the experimenter ensures that there
are captured images for the entire deformation process. This selection also indicates that the single
sensor camera have more than enough memory capacity, but they will lack frame rate (and
resolution). The single sensor in situ cameras will have sufficient frame rate but only suitable for
FRP testing with very accurate triggering.
The last but also in many cases the most important parameter is the price. In many cases the
experimenter cannot just select a camera with the correct specifications, but must use what is
available. These guidelines can be used backward to estimate the strain increment per frame from
a frame rate and recording length.
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A) Minimum feasible number of images for different
failure strain and DIC noise floor
B) Required frame rate as function of the average
strain rate over the specimen.
Figure 9.2 Recording length and frame rate as function of the DIC noise floor. The DIC noise floor can also
be seen as a selected average strain increment over the specimen between each image. The user can select
that there should be x% strain increase between each image and this will set up the requirement for the
camera.
0 1 2 3 4 50
50
100
150
200
250
Failure strain (%)
Req
uir
ed n
um
ber
of f
ram
es
DIC Res.: 0.02%
DIC Res.: 0.03%
DIC Res.: 0.04%
DIC Res.: 0.05%
0 500 1000 1500 20000
2
4
6
8
10
Average strain rate (/s)
Req
uir
ed fr
amer
ate
(Mfp
s)
DIC Res.: 0.02%
DIC Res.: 0.03%
DIC Res.: 0.04%
DIC Res.: 0.05%
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Appendix B Useful conversion 1D wave theory
𝜎𝐼is the incident stress pulse
𝜎𝑅is the reflected stress pulse
𝜎𝑇is the transmitted stress pulse
Figure 9.3 Interface between two slender bars
The transmission and reflection coefficients can be calculated from equation (9.5) and (9.6) and
the equations are plotted in Figure 9.4.
𝜎𝑇
𝜎𝐼
=2𝐴1𝜌2𝐶2
𝐴1𝜌1𝐶1 + 𝐴2𝜌2𝐶2
(9.5)
𝜎𝑅
𝜎𝐼
=𝐴1𝜌1𝐶1 − 𝐴2𝜌2𝐶2
𝐴2𝜌2𝐶2 + 𝐴1𝜌1𝐶1
(9.6)
I
T
Interface
A , A ,
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A) Variation of the mechanical impedance B) Variation of the cross sectional area when the
ZA=ZB
Figure 9.4 Reflection and transmission coefficients as functions the relative mechanical impedance and the
relative cross sectional area.
ZB/Z
A
T/
I
R/
I
X: 3
Y: 0.5
AB/A
A
T/
I
R/
I
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Appendix C Simulation of choked flow
A relationship P(x,t) is sough for compressible air where P is the pressure in an air chamber and x
is the displacement of a piston rod which changes the volume of the air chamber. T is time of the
process. The air chamber is ventilated and air will escape from the chamber as P changes. The
situation is sketched in Figure 9.5 where the pistonrod initial is in rest and then moves forward
with velocity v. The pressure in the air chamber is P1 and the ambient pressure is P2 while the
volume of the chamber is V.
Figure 9.5 Sketch of cylinder with a piston which moves forward at a velocity
v. As the piston moves forward the air is compressed and forced out of the
opening.
The initial state of the air in the cylinder is given by the ideal gas equation for x = 0
𝑃1𝑉 = 𝑛𝑅𝑇 (9.7)
With n as the amount of mol air calculated from air mass (m) and the molar mass (M)
𝑛 =𝑚
𝑀 (9.8)
R is the gas constant and T is air temperature. As the piston rod moves forward the air will be
compressed and escape out of any available holes. The discharge rate of mass/s which leaves the
cylinder with constant velocity and density can be calculated from [109]
�̇� = 𝜌𝑣𝑎𝐴 (9.9)
However the escape velocity depends on the pressure ratio P2/P1 and the density of the air. When
P2/P1 < 0.528 the air has reached sonic velocity at the outlet and the air velocity cannot be
increased further; However the density of the escaping air will increase at this condition with
increasing P1 and the mass rate will increase without an increase in air velocity. This condition is
called a “choked flow” as the air velocity is choked. The choked condition is obtained at a pressure
ratio P2/P1 = 0.528. The hole(s) in the air chamber can be assumed to behave as an orifice and
assuming an adiabatic compression of the air, the instantaneous mass discharge rate for an orifice
can be calculated from
a scaping air
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�̇� = 𝐴√2𝑃1𝜌0 (𝛾
𝛾 + 1) [(
𝑃2
𝑃1
)
2𝛾
− (𝑃2
𝑃1
)
𝛾+1𝛾
] (9.10)
ρ0 is the air density at initial pressure before the process is started and λ is the heat capacity ratio of
air. The instantaneous mass discharge rate for the orifice choked condition is calculated from
�̇� = 𝐴√𝛾𝑃1𝜌0 (2
𝛾 + 1)
𝛾+1𝛾−1
(9.11)
In the case the where air is flowing out of the chamber the pressure ratio P2/P1 will always be less
than 1, however a condition with a P2/P1<1 can be obtained in two ways. The first way is by
maintaining the ambient pressure P2 while P1 is increased. This corresponds to a compression of
the air in the cylinder with constant ambient pressure. The second way is to maintain P1 and lower
P2 towards vacuum. These two conditions create very different variation in the mass flow rate with
the pressure ratio as the choked flow only depends on P1. Figure 9.6 shows the two conditions;
Figure 9.6Mass discharge rate calculated from
equation (9.9) and (9.10) respectivily as function of the
pressure ratio. Increasing P1 with P2 constant is the
situation with the cylinder compression the air.
Decreasing P2 with P1 constant is the situation where
the ambient pressure is lowered and the cylinder
pressure is kept constant.
The condition with P2 going towards vacuum will reach stagnation in the mass flow in the choked
condition whereas the condition with increasing P1 will have continuous and accelerating mass
flow as the pressure ratio is lowered.
A slack adapter placed in lab conditions corresponds to the condition with increasing P1 as the
slack adapter moves and P2 is the constant as the volume in the lab is so big that the air mass
forced out of the slack adapter will not affect the pressure in the lab.
The relationship P1(x) is established by consider the changing volume of the chamber with x. If the
air in the chamber and surroundings are at the same pressure there will be no net flow in and out of
the chamber and the initial mass of air can be calculated from
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1x 10
-3
Pressure ratio (P2/P1)
Mas
s d
isch
arge
rat
e (k
g/s)
Increasing P1 with P2 constant
Decreasing P2 with P1 constant
P2/P1 = 0.528
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𝑚𝑖 =𝑃𝑖𝑉𝑖𝑀
𝑅𝑇 (9.12)
As the piston moves forward the volume the air can occupy is decreased and P1 will increase as P
and V are inversely proportional according to
𝑃 =𝑚𝑅𝑇
𝑀𝑉 (9.13)
However at the moment P1 increases there will be a net flow of air out of the chamber and the flow
is calculated from either equation (9.10) and (9.11) dependent on the pressure ratio P2/P1 so the
chamber pressure cannot solely be calculated from equation (9.7) as the air mass is not constant.
Since the air mass and pressure is related to each other, the chamber pressure must be solved in an
iterative way. The algorithm for solving P(x,t) is
Step 0: Calculate initial air mass from equation (9.12) for t = 0 and x = 0.
Step 1: Increment t (dt) and x (dx) according to the velocity profile of the piston rod and
calculate the new volume V.
Step 2: Calculate a new pressure P1 from equation (9.13) assuming no change in air
mass
Step 3: Calculate the pressure ratio P2/P1 and calculate the mass flow rate from either
equation (9.10) or (9.11) dependent if P2/P1 is higher or lower than 0.528.
Step 4: Update the air mass with 𝑚 = 𝑚 − �̇�𝑑𝑡
Step 5: Update the pressure P1 from equation (9.13) with the update mass.
Step 1 to 5 is repeated until the piston rod is at rest. It is an underlying assumption that �̇� is
constant during dx and dt.
Figure 9.7A shows the effect of varying the increment size for the x while the time increment is
adjusted so the velocity is constant and the same for all three increments. The increment should be
kept down at around 1mm to maintain a smooth approximation. Figure 9.7B shows comparison for
the pressure development for a constant piston rod velocity and a linear rising velocity up to that
of the constant velocity. The pressure build up with x is higher for the high velocity and the
constant velocity gives higher pressure than the corresponding linear rising velocity. This is due to
the faster decreased volume compared to the mass flow out of the chamber. A2 smaller outlet area
will also result in a faster pressure build, which is seen, directly from equation (9.10) and. (9.11)
where the mass flow rate is proportional to the outlet area A2. This developed algorithm can be
used to simulate the force from inclusion of air in a slack adapter and investigate the effect of this.
The algorithm establish and relationship between the pressure in the air chamber and the time and
displacement increments of the piston rod.
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A) Effect of increment in x for constant and equal
velocity of the piston rod for all increments sizes.
B) Comparison of constant and linear rising velocity
of the piston rod.
Figure 9.7 Simulations with A1 = 1283mm2, A2 = 18.6mm2, λ=1.4 (Air), M = 0.029kg/mol, R 8.315 j/mol*k,
T = 293K (Assuming constant temperature), Chamber length = 0.23m.
0 0.05 0.1 0.15 0.20
5
10
15
20
Pistonrod displacement (m)
Air
pre
ssu
re (
MPa
)
1mm steps
5mm steps
20mm steps
0.1 0.12 0.14 0.16 0.18 0.2 0.220
0.5
1
1.5
2
2.5
3
Pistonrod displacement (m)
Air
pre
ssu
re (
MPa
)
Constant velocity 2 m/s
Constant velocity 7.5 m/s
Constant velocity 20 m/s
Rising velocity up to 2 m/s
Rising velocity up to 7.5 m/s
Rising velocity up to 20 m/s
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Appendix D MatLab model for elasto plastic material
A model of a simple elasto plastic material is presented here. An elasto plastic material will deload
in a non-reversible manner and this has to be represented in the material law. A simple power
hardening law is used here for representing an elasto plastic material. The law is given in equation
(9.14)
𝐹𝑠 = 𝐴𝑠𝜎𝑦 (𝜀
𝜀𝑦
)
𝑛
(9.14)
The non-reversible model is given in Figure 9.9 and the output from the model is given in Figure
9.9. The model which simulates the load train of the high strain rate test machine will make an
increment in time, then calculate a new set of displacement and velocities from the previous
known acceleration. The acceleration is then updated from the force function which will be
evaluated with the new increments in time, displacement and velocity. This is done for each time
step until a certain threshold error has been reached. It means strain will be prescribed to the model
as function of time and thus this non-reversible material law must work with prescribed strain
steps and return the force for each strain increment.
Figure 9.8 Non-reversible model to simulate a elasto-plastic material
during a loading profile with unloading stages.
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A) Strain profile given to the material model. Notice
the unloading parts.
B) tress strain cur e from the model. The “dips” are
the unloading stages corresponding to the unloading
part shown in A.
Figure 9.9 Strain profile and stress strain curve of the non-reversible material model.
0 200 400 600 800 1000 12000
0.2
0.4
0.6
0.8
1
1.2
Time (s)
Stra
in (%
)
0 0.2 0.4 0.6 0.8 1 1.20
100
200
300
400
500
Strain (%)
Stre
ss (M
Pa)
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Appendix E Strain gage amplifier
The most demanding case for the strain gage amplifier is the quarter bridge setup (Wheatstone
bridge) which provide the least amount of output signal to the amplifier. The output voltage “e” of
a quarter bridge setup with amplification “Amp”, gage factor “ ”, supply voltage and a applied
strain ε is given as [110]
𝑒 = 𝐴𝑚𝑝1
4𝐾𝐸𝜀 (9.15)
Especially for polymer composite materials the supply voltage must be kept low to minimise local
heating of the specimen when the amplifier is connected before testing the specimen. Due to the
transient nature of the test, the effect of heating the specimen during is the test can be neglected,
especially for metallic materials as they will suffer from a very high internal heating during the
straining of the specimen. For polymer composites the supply voltage should be kept at around 2
volt [111] and this value is used as reference for the next calculations. Exchanging ε with 𝜀̇ in
equation (9.15) will yield
�̇� = 𝐴𝑚𝑝1
4𝐾𝐸𝜀̇ (9.16)
the output rate of the quarter bridge setup and shows there is a linear dependency between the
strain rate imposed to the strain gage and the output rate of the system. The only unknown factor is
the amplification. To maximise the resolution a strain signal should always be amplified to the
maximum limit of the DAQ system. In many cases this is ±10 volt. This is used as a reference
value. The required amplification will then depend on the maximum strain to measure and the
amplification is calculated from
𝐴𝑚𝑝 = 4𝑒
𝐸𝐾𝜀𝑚𝑎𝑥
(9.17)
This relationship is shown in Figure 9.10A where a maximum strain of 1% will require a
amplification of 1000, 5% will required 200 and 10% maximum strain will require 100 in
amplification. The output rate is shown as function of strain rate in Figure 9.10B.
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A) Amplification factor as function of maximum strain
to measure. Max output voltage e= 10V. Max supply
voltage E=2V. Gage factor K = 2.
B) Output rate as function of the strain rate
imposed to the system. The selected amplifier must
have a slewrate which is above the curves to avoid
any damping of the amplified signal
Figure 9.10 Selection curves for high strain rate strain gage amplifiers.
0 2 4 6 8 100
2000
4000
6000
8000
10000
Maximum strain (%)
Am
plif
icat
ion
fact
or
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
_"
V=7s
1% Maximum strain
5% Maximum strain
10% Maximum strain
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Appendix F Parameters for simulations in Chapter 2
%% -- Model parameters for classical slack adapter system % -- Areas K.Ap = 4778/1e6; % Pistonrod cross sectional area (m2) K.Ar = 1200/1e6; % Pressure area of the rubber damper (m2) K.As = 5*15/1e6; % Slackadapter crosssectional area (m2) K.Asp = 1*15/1e6; % Specimen crosssectional area (m2) % -- Masses K.M1 = 1.83; % Mass at node 1 K.M2 = 1.83+0.64; % Mass at node 2 K.M3 = 0.64+0.104; % Mass at node 3 K.M4 = 33; % Mass at node 4 Piston rod % -- Lengths K.Lr = 0.003; % Rubber damper thickness (m) K.Ls = 0.15; % Slackadapter length (m) K.Lsp = 0.05; % Specimen free length (m) K.Li = 0.180; % Free acceleration length in slack adapter % -- Stiffness K.Es = 42e9; % Slackadapter stiffness (Pa) K.Klc = 26e9; % Loadcell stiffness (N/m) K.Esp = 42e9; % Specimen stiffness (Pa) % -- Other material constants K.Vi = 0; % Initial velocity of node 1 (m/s) K.Vn = 20; % Rated velocity K.Pn = 270*1e5; % Pressure drop at initial velocity (Pa) K.C1 = 50e6/2; % Rubber material constant neo Hookian law (Pa) K.EpsF = 0.025; % Failure strain of specimen K.SigY = 800e6; % Yield stress K.Epsy = K.SigY/K.Esp; % Yield strain K.n = 0.1; % Work hardening exponent K.Steel = 0; % Type of specimen 0 = Linear elastic, 1 = Elastic plastic
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